1) HALACHAH: "SHINUY MAKOM"
OPINIONS: The Gemara discusses whether or not a person is required to recite a new Berachah when he began to eat in one place and then moved elsewhere. This subject is a matter of dispute among the Tana'im and Amora'im. According to Rebbi Yehudah, one must recite a new Berachah in the second place. The Rabanan, though, maintain that one needs to recite a new Berachah only for foods which do not normally require a Berachah Acharonah in the place they were eaten. When one eats foods that do require a Berachah Acharonah in the original place, one does not need to recite a new Berachah in the new place. Rav Chisda rules like the Rabanan.
The RASHBAM explains the reasoning of the Rabanan as follows. When one eats a food that requires a Berachah Acharonah in its place, the recitation of the Berachah Acharonah marks the end of the meal. As long as the Berachah Acharonah has not been said, any subsequent eating is a continuation of that meal. When the person does not recite the Berachah Acharonah in its place, this is an indication of his intent to continue his meal elsewhere. Therefore, he does not recite a new Berachah Rishonah in the new place.
When one eats a food that does not require a Berachah Acharonah in its place, getting up and leaving indicates that the person has finished and does not intend to eat any more. The fact that he has not said a Berachah Acharonah in the first place is insignificant, as he will recite it elsewhere. Therefore, his leaving indicates his intent to stop eating, and any subsequent eating in another location will be a new session and require a new Berachah Rishonah. (See, however, the CHIDUSHEI HA'RAN (DH Lo Shanu), who understands the Gemara differently.)
What is the Halachah?
(a) The RASHBAM and the ROSH (10:6) rule like Rav Chisda because the Gemara itself states that a Beraisa supports his opinion. Furthermore, his opinion is that of the majority (the Rabanan).
Tosfos and the Rosh add that if a person does go from one place to another while he eats an item which does not require a Berachah Acharonah in its place (and thus he needs to recite a Berachah Rishonah in the new location), he does not need to recite a Berachah Acharonah for the food he ate in the original place. Shinuy Makom is considered a "Hesech ha'Da'as" (an interruption) which warrants a new Berachah Rishonah; it is not considered an "Akirah" (a termination) of his first meal. When he eats the food in a new place, he does not begin a new meal; rather, he continues the original meal. It is one meal with an interruption, and not two separate meals.
(b) However, the RIF rules like Rav Sheshes, that one must always recite a new Berachah, because Rebbi Yehudah, in the earlier Beraisa, clearly supports the view of Rav Sheshes. Even though the Rabanan argue with Rebbi Yehudah, their opinion is not stated explicitly (it is only inferred). The Rif does not rely on the Gemara's assertion that there are some Rabanan who argue with Rebbi Yehudah.
This is also the opinion of the RAMBAM (Hilchos Berachos 4:4). In fact, the Rambam is also stringent with regard to a Berachah Acharonah. That is, if someone goes to a new place to eat, not only does he need to recite a new Berachah Rishonah there, but he also needs to recite a Berachah Acharonah for what he ate in the first location. The Rambam views a Shinuy Makom as an "Akirah," a total termination of his first meal.
HALACHAH: The SHULCHAN ARUCH (OC 178:1) rules like the RAMBAM, who is stringent in both respects (like Rav Sheshes). He requires a Berachah Acharonah for the food one ate in the original place, and a new Berachah Rishonah for the food one wishes to eat in the new place, even for bread. The REMA, however, rules leniently in both respects, like the ROSH.
The MISHNAH BERURAH (OC 178:28), in the name of the MAGEN AVRAHAM, adds that the Halachah that one who ate bread (an item which requires a Berachah Acharonah in its original place) does not need to recite a new Berachah Rishonah applies only when he ate at least a k'Zayis of bread in the original place.
All agree, however, that l'Chatchilah a person should not change his location while he eats, unless he needs to perform a Mitzvah or he had in mind to change his place when he recited his original Berachah. According to the SHULCHAN ARUCH, one who had in mind to change his place may go from one room to another in one house. According to the REMA (who rules like Rav Chisda and does not require a new Berachah when one eats bread), one may have in mind to travel even from one house to another (MISHNAH BERURAH OC 178:33 and 178:40).
RAV MOSHE FEINSTEIN (Igros Moshe OC 2:57), who follows the lenient ruling of the Rema, adds that even if one eats fruit (or other items that normally require a new Berachah if one changes his place), he does not need to recite a new Berachah in the new location if he continues to eat as he goes to the new place and does not stop eating for a period of "Kedei Dibur."
It should be noted that when the REMA rules like Rav Chisda, he rules that b'Di'eved one does not need a new Berachah Rishonah in the new location. What is the Halachah, though, when a person needs to leave his place for the sake of a Mitzvah and he plans to return? Is it preferable that he recite a Berachah Acharonah before he leaves, and begin his meal anew when he returns? Or should he not recite a Berachah Acharonah until after he returns and finishes his meal?
The BI'UR HALACHAH concludes that it is better not to recite a Berachah Acharonah when one intends to return and eat more, in order to avoid unnecessary blessings. Rather, one should leave without a Berachah Acharonah, unless there is a possibility that he will be away for a long time (more than 72 minutes), after which he will not be able to recite the Berachah Acharonah. In such a case, he should recite the Berachah Acharonah before he leaves (Bi'ur Halachah, end of DH b'Lo Berachah).
2) FOOD ITEMS THAT NEED A "BERACHAH ACHARONAH BI'MEKOMAN"
OPINIONS: Rav Chisda says that if a person changes location while he eats a food that requires a Berachah Acharonah in the place it was eaten, he does not need to recite a new Berachah Rishonah in the new location. What foods require a Berachah Acharonah in the place where they were eaten ("Te'unin Berachah Achareihen bi'Mekoman")?
This question is relevant in practice in two important ways. First, for which foods does a person need to recite a Berachah Acharonah in the place where he ate them? Second, for which foods does a person need to recite a new Berachah Rishonah if he changed places?
(a) The RASHBAM writes that this requirement applies to all foods of the Shiv'as ha'Minim, the seven species of produce of Eretz Yisrael. This is also the opinion of the RAMBAM (Hilchos Berachos 4:1).
(b) TOSFOS (101b) and the ROSH (10:6) write that this refers only to foods made from grain.
(c) The RASHBA (Berachos 53b) rules that only bread requires a Berachah Acharonah in its place.
HALACHAH: The SHULCHAN ARUCH (OC 178:5) cites the opinion of the Rambam (a) and the opinion of Tosfos and the Rosh (b). The REMA quotes the Rashba (c). The VILNA GA'ON rules like Tosfos that bread and other grain-based foods require a Berachah Acharonah in the original place.
The MISHNAH BERURAH (178:45), therefore, rules that a person should be stringent and, l'Chatchilah, recite a Berachah Acharonah in the original place whenever he eats any food of the Shiv'as ha'Minim. On the other hand, one should follow the ruling of the VILNA GA'ON and not leave the place where he started to eat a food of the Shiv'as ha'Minim that is not grain-based, because he might be required to recite a new Berachah Rishonah if he goes to a new place (since, according to the Vilna Ga'on, such foods are not items that require a Berachah Acharonah in their original place).