Gemara

(R. Avin citing R. Yochanan): If Reuven was laden with food and drink, and went back and forth between Reshus ha'Rabim and Reshus ha'Yachid all day, he is exempt until he stops.

(Abaye): He is not liable until he stops to rest.

He learns from Rabah;

(Rabah): If one carried four Amos in Reshus ha'Rabim, but stopped in the middle to rest, he is exempt.

If he stopped to adjust, he is liable (when he finishes).

If after four Amos he stopped to rest, he is liable. If he stopped to adjust, he is exempt.

153a (Mishnah): If one was on the road when Shabbos was about to come, he may give his wallet to a Nochri.

If no Nochri is around, he puts it on his donkey.

(Rava): This is permitted only for his wallet, but not for a found object.

If a donkey, minor, lunatic and Cheresh (deaf person) are all available, he should put it on the donkey, and not one of the others.

153b (R. Yitzchak): Chachamim had another Heter, but they did not want to reveal it.

He carries it less than four Amos at a time. They didn't want to reveal it, lest he [accidentally] carry four Amos at once in Reshus ha'Rabim.

(Rav Ada bar Ahavah): If [on Shabbos] one was carrying a load on his shoulder, he may run to his house;

It is permitted only if he runs. He may not walk slowly.

This is because [if he walks,] he has no Heker (reminder that it is Shabbos), and he may inadvertently [stop, and thereby] do Akirah and Hanachah.

Question: When he reaches his house, he must stop momentarily (e.g. to open the door. Afterwards,) he enters from Reshus ha'Rabim to Reshus ha'Yachid!

Answer: He throws it into the house in an awkward way.

Rishonim

Rambam (Hilchos Shabbos 13:8): If one had an item in his hand or on his back, and went back and forth between Reshus ha'Rabim and Reshus ha'Yachid, even if he did so all day until night, he is exempt, for he did Akirah but not Hanachah. Even if he stopped in the middle to adjust the load, he is exempt, until he stops to rest is liable. If he stopped to adjust, he is exempt.

Rambam (9): If one's load was on his shoulder and he ran with it even the entire day, he is not liable until he rests. This is if he was running. If he walked slowly, this is like Akirah and Hanachah, and it is forbidden. Therefore, if one was carrying his load when Shabbos came, he runs with it until getting home, and casts it down awkwardly.

Magid Mishneh: He may not walk slowly, lest he come to do Akirah and Hanachah, i.e. he will rest. As long as he does not rest, he is exempt, for walking is unlike stopping. The Rambam says that walking slowly is like Akirah and Hanachah, and it is forbidden. He did not say that one is liable. This is obvious! The Ramban says that if one was carrying his wallet when Shabbos came, he may run with it and do just like with his load.

Ran (66b DH v'Havi, citing the Ramban): We permit only with a load, since it is abnormal to run with it. Therefore, one who runs with it has a Heker. One who carries a wallet has no Heker. If you would not say so, why didn't the Mishnah give this solution for a wallet? If you will say that Chachamim did not want to reveal it, it should have been taught with (the other solution that they did not want to reveal, i.e.) carrying less than four Amos at a time!

Shiltei ha'Giborim (Sof 66a): He must begin running during the day, so there is no Akirah on Shabbos (Sefer ha'Terumah). Since we do not have Reshus ha'Rabim, only Karmelis, perhaps it is permitted in every case. Sefer ha'Terumah connotes that if he did Akirah, it is forbidden. This is difficult, for we hold that Akirah without Hanachah is nothing!

Kaf ha'Chayim (OC 266:49): The Magen Avraham (18, brought below) permits running with a wallet when there is no other solution. This implies that in any case one may run with a load. Shiltei ha'Giborim's Perush of the Ramban connotes like this; this is wrong (Eliyahu Rabah).

Poskim

Shulchan Aruch (OC 266:8): Some say that the Heterim are only for one who thought that he had time and it became dark on the road. If one left his house close to dark and forgot and took out into Reshus ha'Rabim, they did not permit any of these solutions.

Source: R. Yerucham, cited in Beis Yosef DH Kasav Rabbeinu.

Shulchan Aruch (11): If one's load was on his shoulder and Shabbos came, he runs with it runs with it until getting home. He must run. He may not walk, for there is no Heker, so he is prone to do Akirah and Hanachah. Sometimes he stands (still), unaware of this. When he runs, there is a Heker.

Beis Yosef (DH Haysah): Rashi explains that we explain he was carrying the load from before Shabbos.

Magen Avraham (16): If he stopped to adjust the load he is exempt.

Mishnah Berurah (30): This is not called Hanachah.

Mishnah Berurah (29): This is only when there is no Nochri, animal, Cheresh, lunatic or child, and in a place where he cannot take it less than four Amos at a time, e.g. he fears lest robbers take it, or there is water wider than four Amos. Since we do not have a full Reshus ha'Rabim, perhaps it is permitted in every case.

Shulchan Shlomo (12, cited in Dirshu edition of Mishnah Berurah): The Mishnah Berurah means that in any case one may run. I say that he need not run

Chazon Ish (OC 103:19): If one carries from one Reshus ha'Yachid to with Karmelis in the middle, this is Shevus d'Shevus (a mid'Rabanan fence for an Isur mid'Rabanan), so one may be lenient for the need of a Mitzvah, and all the more so Bein ha'Shemashos. Therefore, if one was in the Beis ha'Keneses or train when it became dark, and he has something that he cannot leave there for it is not guarded, he may walk to his house without resting in the middle.

Kaf ha'Chayim (55): Since we do not have Reshus ha'Rabim, it is permitted even if he (had stopped, and) began running after dark, as long as he will cast it awkwardly. There is Akirah without Hanachah.

Shulchan Aruch (ibid.): When he gets home, in order not to stand for a little and take from Reshus ha'Rabim to Reshus ha'Yachid, he casts it awkwardly, i.e. abnormally, e.g. from his shoulders and in back.

Mishnah Berurah (32): The Roke'ach explains that he tilts his back towards the Chatzer and casts it from his shoulders. There is no Isur mid'Oraisa when acting awkwardly. To avoid loss of his bundle, Chachamim permitted this.

Bi'ur Halachah (DH Ki): The Gemara asked about when he reaches his house, he must stop momentarily, and answered that he casts it awkwardly. The Gemara connotes that since he will cast it awkwardly, he may do Hanachah close to his house, since there was no Akirah at the beginning, for he was loaded from before Shabbos. (Even though he did Akirah after getting close to his house, he will cast it down awkwardly, so there is not a proper Hanachah - PF.) Eliyahu Rabah brings so from Ma'adanei Yom Tov. In any case some are strict not to rest. The Gemara said that he must stop a little. He does not truly stop, just it looks like he stops.

Shulchan Aruch (12): Some permit this only for a bundle, but not for a wallet. Some say that the same applies to a wallet.

Beis Yosef (DH Kasav Sham): The Magid Mishneh says unlike the Ran, that the Ramban permits even a wallet, and so says Hagahos Maimoniyos in the name of Semag. Semak and Sefer ha'Terumah say so. R. Yerucham brought both opinions, and did not decide between them.

Magen Avraham (17): Running with a wallet is not a Heker, for it is light.

Gra (DH Yesh and DH v'Yesh): The first opinion forbids, since the Gemara did not mention this Heter with a wallet. The latter opinion holds that it was not mentioned, for it is forbidden when there are other solutions.

Mishnah Berurah (35): If through untying his belt he will suffer a loss, perhaps also the Gra would permit (carrying it to a hiding place).

Bi'ur Halachah (DH d'Hu): Seemingly, the lenient opinion is unlike the opinion in Sa'if 8 (which forbids one who left close to dark). Here we permit running, which is the worst solution! However, this is wrong. If so, why did the Mechaber in Sa'if 8 bring only the stringent opinion? Rather, even the lenient opinion here agrees with Sa'if 8. We are lenient only for one who was close to Ones, but not for one who close to negligence. This is why the Rema is stringent. The Gra connotes like this. This answers the Taz' question (below). Nehar Shalom says so.

Kaf ha'Chayim (57): One may not be lenient about something he finds, which was not yet his.

Magen Avraham (18): If he has a donkey but fears lest robbers take it from his donkey, he may run.

Rema: If one forgot and left his wallet on him on Shabbos, if he is in his house, he may walk to a room to untie his belt and it will fall there and he will hide it. If he is in the market, he may not bring it home. Rather, he unties his belt in the market, and it falls. He tells a Nochri to guard it. We are not concerned if the Nochri brings it.

Taz (6): Why does the Rema say this? The Mechaber said that one opinion permits this! There is no proof from the Agudah, who is the author of this law. He surely holds like the first opinion, which forbids! Since this is a mid'Rabanan law, one who relies on the lenient opinion does not lose.

Kaf ha'Chayim (56): The primary opinion is lenient, like the Taz.

Magen Avraham (20): The same applies where there is no Eruv. If he fears lest Nochrim take it, he may run with it.

Mishnah Berurah (36): This Heter is only if he remembers while walking and he did not stop to rest, so he did not do an Akirah when he began running. If he fears to rely on the Nochri, he may command him to take it to the Yisrael's house. When possible, this is better than all the other ways mentioned above.

Kaf ha'Chayim (61): We are lenient to command the Nochri, for the main opinion is that nowadays there is no Reshus ha'Rabim, so this is Shevus d'Shevus in order to avoid a loss, which is permitted. It is better to tell the Nochri to carry through one of the methods permitted for a Yisrael.