Gemara

17b: When R. Zeira came to Eretz Yisrael, he ate animals slaughtered in other rings, which Rav and Shmuel disqualify.

Rabanan: You come from Bavel, where people conduct like Rav and Shmuel!

R. Zeira: Only Rav Yosef says that they disqualified this.

Question: Does R. Zeira hold that a visitor need not adopt the stringencies of his home town and the place he visits?!

Answer #1 (Abaye): That applies only to one who travels within Bavel, or within Eretz Yisrael, or goes from Eretz Yisrael to Bavel;

If one goes from Bavel to Eretz Yisrael, since we (in Bavel) are subordinate to them, he conducts in all ways like a Ben Eretz Yisrael.

Answer #2 (Rav Ashi): Even if a Ben Bavel goes to Eretz Yisrael, if he does not intend to return, he adopts all the customs of his new locale;

R. Zeira intended to remain in Eretz Yisrael.

Pesachim 51a: When Rabah bar bar Chanah came to Bavel, he ate Chelev De'aisra (on the bent side of the stomach. Bnei Eretz Yisrael eat it, but Bnei Bavel do not. Rav Avira Sava and Rabah bar Rav Huna came to see him. He covered it up.

Abaye (to Rav Avira and Rabah): He treated you like Kusim (he did not want you to see that it is permitted)!

Question (Mishnah): He must be stringent like his old and new residences.

Answer #1 (Abaye): That is within Eretz Yisrael, or within Bavel, or from Bavel to Eretz Yisrael, but not from Eretz Yisrael to Bavel.

Answer #2 (Rav Ashi): We can even say that one who goes from Eretz Yisrael to Bavel gets both stringencies, i.e. if he intends to stay there. Rabah intended to return to Eretz Yisrael.

51b (Mishnah): One who goes from a place (where people work on Erev Pesach...or vice-versa... he may not deviate...)

Question: Granted, if they work in his origin but not in the new place, he must be stringent like the new place; and he may not deviate due to strife, so he does not work;

However, if they did not work in his origin and in the new place they do, since he may not deviate, he must work. However, he must follow the stringencies of both places!

Answer #1 (Abaye): 'He may not deviate' refers only to the Reisha (in his origin, they work).

Answer #2 (Rava): It applies to the Seifa. Deviating through not working will not cause strife. People who see him will not say that he forbids working. Rather, they will that he is like many other idle people.

Rishonim

Rosh (Pesachim 4:4): Rabah bar bar Chanah was lenient only in private, due to strife. Even one who goes to a place where they are lenient should act leniently and not be stringent in front of them in a matter that would be evident, due to strife. However, Melachah is not evident, for they are many idle people. If not for this, we would tell him to violate his custom and do Melachah with them, since there is no Isur Torah, and Shalom is so important!

Ran (Pesachim 17b DH v'Rabah): Whether Bnei Bavel forbade Chelev De'aisra from letter of the law or for a stringency, we ask why Rabah bar bar Chanah was not stringent like his new locale. Anyone who intends to stay acts like them, to be lenient or stringent. Abaye answered that Bnei Eretz Yisrael need not follow stringencies of Bnei Bavel. Rav Ashi said that he intended to return, therefore letter of the law he was permitted. He ate in private, to avoid strife. He had to cover it up, due to strife. Why did Abaye say 'he treated you like Kusim'? He ate in private. He did not need to expect that they will enter and see him. There is no strife among Chachamim, who know that he acts properly.

Tosfos (18b DH Yosef): R. Zeira said that Rav Yosef never heard from Rav and Shmuel. It was a mere stringency that they did not eat it there, not based on Rav and Shmuel. The Gemara challenged R. Zeira, for even if Rav and Shmuel did not say so, one must be stringent like the local custom.

Tosfos (51a DH Rabah): In Chulin, Rav Ashi answered that R. Zeira did not intend to return. This is slightly difficult. One keeps the stringencies of his origin (only) when he intends to return, and the stringencies of his new place (only) when he will settle there. The Mishnah is l'Tzedadin (there is no one case in which he keeps both stringencies, unlike the simple reading of the Mishnah).

The questioner (in Teshuvas ha'Rashba 1:337): R. Yehonason explained like this. The Ramban says that the Mishnah discusses one who intends to return. He gets stringencies of the new place regarding permitted things that they forbid for a fence, like the case of Bnei Beishan (they would not to go from Tzur to Tzidon on Erev Shabbos). In such case, we are concerned for strife.

Rashba: Presumably, the Mishnah discusses both fences and matters that they believe are forbidden letter of the law. Bnei Bavel forbid Chelev d'Eisra due to letter of the law, and the Gemara asked why Rabah ate it. It is difficult to say that the Mishnah is li'Tzedadin. Rather, if one goes from Bavel to Eretz Yisrael and intends to return, he gets both stringencies. Anyone who intends to return is like the people of his origin. If he is in a place more esteemed than his origin, e.g. Eretz Yisrael, he also gets the local stringencies, like Abaye said. Rav Ashi agrees. He merely teaches that anyone who intends to return keeps his old stringencies. Even if one travels within Eretz Yisrael or within Bavel, and intends to return, he gets both stringencies. Since the new place is as esteemed as his origin, he may not be lenient in front of them. R. Yanai would not move a lamp (on Shabbos) in R. Yochanan's region. Rami bar Tamri traveled within Bavel, and had he entered the Techum, he would have had to keep the local stringencies, for their honor. Presumably, he intended to return. If not, why did Rav Kahana challenge Rami? Rav Kahana did not know whether Rami intends to return! Most people intend to return! However, one who goes from Eretz Yisrael to Bavel with intent to return does not get the stringencies of Bavel. He may be lenient in front of them. We say (Yevamos 14) that Beis Shamai did not act unlike Beis Hillel in front of them, but Beis Hillel acted unlike Beis Shamai in front of them. Abaye said so, and Rav Ashi merely qualified that this is only if he intends to return.

Poskim

Shulchan Aruch (OC 496:3): A Ben Eretz Yisrael who came to Chutz la'Aretz may not do Melachah on Yom Tov Sheni in a Yishuv (settled area), even if he intends to return. He is permitted until he reaches a Yishuv, even if he does not intend to return, for he was not yet fixed to be like them.

Mishnah Berurah (13): If a Ben Chutz la'Aretz came to Eretz Yisrael, if he intends to return, he must observe Yom Tov Sheni. He must pray Tefilah of Yom Tov in his house, in private. Acharonim say that one who moves with his wife and children to another place for business, even though when he moved he intends to return, he is like one who does not intends to return. Since he uprooted his family, presumably if he finds income in his new place, he will not leave there.

Kaf ha'Chayim (38): If a Ben Chutz la'Aretz came to Eretz Yisrael and he intends to return, he keeps Yom Tov Sheni properly.

Kaf ha'Chayim (39): Hilchos Ketanos (1:4) says that if there are less than 10 married men from Chutz la'Aretz, they should not make their own Minyan. It is improper to take a Ben Eretz Yisrael to complete the Minyan. Mateh Asher and Sama d'Chayei permit. Some permit, and some forbid. Pe'as ha'Shulchan says that until the year 5481 Rabanan forbade, and then they decided to permit.

Kaf ha'Chayim (40): Hilchos Ketanos says that if a single man came to Eretz Yisrael, even if he intends to return, he keeps only one day, for if he would find a wife he would stay. If he relies on his parents (financially), he keeps two days. Chayim She'al agreed. Ginas Veradim (4:14) says that he always keep two days if he intends to return, but a single Ben Eretz Yisrael in Chutz la'Aretz keeps only one day if he intends to return, even if he does not depend on others.

Kaf ha'Chayim (42): If a Ben Chutz la'Aretz came to Eretz Yisrael with intent to remain, but his wife is still in Chutz la'Aretz, the custom in Yerushalayim is to keep two days, for perhaps he will return.

Kaf ha'Chayim (43): If a Ben Chutz la'Aretz came to live in Eretz Yisrael, and if he finds a place he will send for his wife, and if she refuses he will go to bring her without delay, since she wants to come, and even if she refuses he will leave her, since he resolved to stay, he is like a Ben Eretz Yisrael regarding Tefilah and similar matters. However, he should not eat Chametz on the eighth day of Pesach.

Avkas Rochel (26): If a Ben Chutz la'Aretz came to Eretz Yisrael and he intends to return, he keeps Yom Tov Sheni properly. People regularly come to Eretz Yisrael for festivals. They even make a Minyan for Yom Tov Sheni like in Chutz la'Aretz, and read the Haftorah with a Berachah. Gedolim saw, and never questioned this. Why aren't we concerned for strife? They could pray a weekday Shemoneh Esre in private! We must say that concern for strife applies only to Melachah or similar matters, which could lead to transgression. There is no such concern if one will pray like his origin.

Kaf ha'Chayim (46, citing Birkei Yosef 4): Maharikash (90) permits a Ben Chutz la'Aretz in Eretz Yisrael to tell a Ben Eretz Yisrael to do Melachah for him on Yom Tov Sheni. Zera Avraham agreed, but Ginas Veradim (4:17), Mahari Molko and others forbid.

Ha'Gaon R. S. Z. Auerbach, Ztz"l permits. Ha'Gaon R. Y.S. Elyashiv, Shlita permits only for Mitzvos, for there is no Isur to discuss doing such Melachah. (Brought in Yom Tov Sheni k'Hilchasah, Teshuvos 2:3 and 3:1.)

Shulchan Aruch (468:4): If one goes from a place where people do Melachah (on Erev Pesach) to a place where they do not do, he may not do Melachah in the Yishuv, due to strife. He may do in the Midbar (wilderness).

Gra (DH ha'Holech): The Rambam says so. Due to Tosfos' question, the Rambam rejected the Sugya in Chulin, and rules like the Sugya in Pesachim. The Mishnah discusses one who does not intend to return. R. Zeira ate because he argues with Rav Yosef. In the conclusion of Pesachim, we must say like the Yerushalmi, that one need not accept a local stringency based on a mistake. R. Zeira held that it is a mistake.

Bi'ur Halachah (DH ha'Holech): The Magen Avraham holds that Tosfos and the Rosh agree with the Rambam. The Gra and others hold that Tosfos and the Rosh forbid even in the Midbar in this case.

Shulchan Aruch (ibid.): If one goes from a place where they do not do to a place where they do, he may not do. We put on him stringencies of both places. If so, he should not show that he is idle due to Isur. One should never deviate (from the local custom), due to strife. Similarly, one who intends to return to his origin is lenient or stringent like people of his place, but he should not show this to people where he is now, due to strife.

Mishnah Berurah (14): If one goes from a place where they are stringent to a place where they are lenient, if he will settle there, he may be lenient. If he will not settle there, he must be stringent, but he should not show them that he deviates, due to strife. He must be stringent in private.

Mishnah Berurah (23): Anything forbidden letter of the law, even mid'Rabanan, one may not transgress it due to concern for strife. It is better to have strife. One who transgresses mid'Rabanan laws is Chayav Misah b'Yedei Shamayim.