Gemara

Shlomo asked Chachamim: My father is dead, in the sun (the body is prone to decay). What should I do?

Chachamim: Put a loaf or child on top of your father. You may move his body together with the loaf or child.

124a (Mishnah): All Kelim may be moved for a need or without a need;

R. Nechemyah says, they may be moved only for a need.

(Rava): 'For a need' is Kli she'Melachto l'Heter for its use or place. 'Without a need' is from the sun to the shade, or Kli she'Melachto l'Isur for its use or place, but not from the sun to the shade;

R. Nechemyah permits moving even Kli she'Melachto l'Heter only for its use or place, but not from the sun to the shade.

Question (Abaye - Beraisa): If there is garlic in a Meduchah (a bowl in which things are pounded), it may be moved. If not, it may not.

Answer (Rava): That refers to moving it from the sun to the shade.

142b (R. Oshaya): If one forgot a wallet in a Chatzer, he puts a loaf or child on top of it and moves the wallet (with the loaf or child).

Rav Yitzchak and Rav Yehudah bar Shila taught the same Heter regarding a brick forgotten in a Chatzer and a sack of coins forgotten in Reshus ha'Rabim (to carry it less than four Amos at a time, or within a wall of people).

Mar Zutra: The Halachah follows all of these teachings when the Muktzeh was forgotten.

Rav Ashi: It is forbidden even when it was forgotten. The leniency to put a loaf or child on top and carry applies only to a Mes (that might decay in the sun, due to Kavod ha'Briyos).

Abaye would put a spoon on top of sheaves (to permit carrying them). Rava would put a knife on top of a (raw, unsalted) dove.

Rav Yosef: Chachamim only permitted for one who forgot, but not to intend to move this way!

Abaye: Really, I could move the sheaves by themselves, for one could lie on them. I am important, therefore I am stringent to put a spoon on top (lest people learn from me to be lenient).

Rava: Really, I could move the dove by itself, for it could be eaten raw. I am important, therefore I put a knife on top.

Rishonim

Rosh (21:2): The Halachah does not follow R. Oshaya, who the Shulchan Aruch moving a wallet through a loaf or child. Rav Ashi taught this is permitted only for a Mes. R. Nisim [Ga'on] derived that those who rely on a loaf or child to move a lamp that was burning on Shabbos, this is improper.

Teshuvas ha'Rosh (22:8): A Beraisa permits moving a bowl in which things are pounded if there is garlic inside. Rava explains that it permits moving the bowl from the sun to the shade.

Question: Rav Ashi says that the leniency through a loaf or child applies only to a Mes!

Answer (Beis Yosef OC 308 DH Kosav ha'Rosh): Ra'avan says that Rav Ashi discusses intrinsic Muktzeh like rocks (similar to a Mes) or a wallet full of coins, for it is Batel to the coins, which are like rocks. Some permit carrying the bowl from the sun to the shade because it served the garlic before Shabbos. Kol Bo and the Hashlamah forbid.

Rashba (Avodas ha'Kodesh Beis Mo'ed 5:2:16): The only reason a Meduchah was forbidden is because it is used for dicing. Now it is serving Heter [holding garlic] in its Melachah. The very reason to forbid it permits it, just like a pot, which may be moved with the Tavshil (cooked food) inside.

Poskim

Shulchan Aruch (OC 308:5): Some permit to move a Keli she'Melachto l'Isur from the sun to the shade by putting a loaf or child on it (and moving the latter through the Muktzeh).

Taz (4): It seems that one should not be lenient about this. See what I wrote in 311:5 (below).

Magen Avraham (13): The same applies if one put something else in it that one may move.

Gra: They learn from 124a (if there is garlic in a Meduchah, one may move it from the sun to the shade). We say that Chachamim permitted through a loaf [or child, only regarding a Mes] regarding something totally forbidden to move, which is not a Kli. The Rashba disagrees [with the Shulchan Aruch's Heter].

Shulchan Aruch (311:5): If one needs the place where a Mes is, or what the Mes is resting on, one may move it Min ha'Tzad, i.e. he flips it from bed to bed, since it is for the sake of Heter. Chachamim permitted moving only a Mes through a loaf or child, but not other things forbidden to move.

Taz (5): Do not ask from 308:5, which permits moving a Kli she'Melachto l'Isur from the sun to the shade through a loaf or child. Here (a Mes) is different, for it was Huktzeh (dismissed) due to itself, like a rock. There, it is a Kli. We learned that one may move a Meduchah due to garlic inside, even from the sun to the shade. Do not ask from 310:8, in which we permit moving a Kli with Isur (broken pieces of wood) and Heter, even though the Isur is like rocks. There, the Isur is Batel to the Heter. It is as if there is no Isur. The Rosh and Tur concluded here "therefore, those who rely on a loaf or child to move a lamp that was burning on Shabbos, this is improper." Why is this different than 308:5? The lamp is a Kli, just it is Melachto l'Isur! Why does a loaf or child help there more than here? Here is from Piskei ha'Rosh, and there is from Teshuvas ha'Rosh. What he wrote here is primary. The Beis Yosef (279) brought from the Rivash not to move a lamp via a loaf put on it on Shabbos, for Chachamim permitted moving only a Mes through a loaf or child. Therefore, one may not rely on the leniency of 308:5. One could say that there we are more lenient because one may move the Muktzeh itself l'Tzorech Gufo u'Mkomo, and the loaf or child is only to permit from the sun to the shade, whereas the lamp is forbidden even l'Tzorech Gufo u'Mkomo, therefore a loaf or child does not help.

Gra (ibid.): His question is not difficult. A lamp that was burning on Shabbos is Muktzah due to Isur. He overtly forbade moving it [through lighting in it before Shabbos] even l'Tzorech Gufo. It is like a wallet. In any case, the Taz is correct. A Meduchah is different, like the Rashba wrote. However, one may move it only if it has inside something special to it (normally mashed in it). The Beraisa connotes like this. It permits specifically when there is garlic inside. The Rashba wrote like this in his Chidushim (123a DH Ha).

Mishbetzos Zahav (4): A lamp was Muktzeh Bein ha'Shemashos due to the oil inside. It is forbidden even l'Tzorech Gufo u'Mkomo, even after it extinguishes. The stringent opinion permits a Kli only with what it is special for, so a loaf or Peros would not permit a pot. According to the lenient opinion, does a loaf or child permit Muktzeh due to of Chisaron Kis (something valuable that one uses only for its special use)? Perhaps it is better than rocks (which they do not permit), for it is a Kli. This requires investigation.

R. Akiva Eiger (on 308:5): One may move Muktzeh due to Chisaron Kis through a loaf or child, even though one may not move it by itself l'Tzorech Gufo u'Mkomo. The Gemara (123a) wanted to learn from the Beraisa of a Meduchah that one may not move Kli she'Melachto l'Isur l'Tzorech Gufo u'Mkomo. Even though one may not move it by itself [in the Hava Amina], one may move it along with what is in it. The same applies to Muktzeh due to Chisaron Kis. If so, a loaf or child permits moving a lamp that burned on Shabbos. The Ba'al ha'Ma'or (47b) brought so in the name of a Chacham.

Note: I did not see anyone mention that R. Akiva Eiger contradicts what he wrote in the following Teshuvah. Kaf ha'Chayim (55) cites only the Teshuvah.

R. Akiva Eiger (Teshuvah 1:22): One may not move Muktzeh due to Chisaron Kis through a loaf or child, since it has no Heter to move it (by itself), just like rocks. However, one may move a Mohel's knife (after using it on Shabbos) through a loaf or child. This is because as long as the knife is in his hand, one may move it to where he wants (Rema 266:2). Tosefes Shabbos says that even though this Heter is only for Kli she'Melachto l'Isur, which one may move l'Tzorech Gufo u'Mkomo, but not for rocks and coins (also Even ha'Ozer says so, unlike the Magen Avraham), a Mohel's knife is better, since it is permitted for the Mitzvah of Milah. It is like a Kli she'Melachto l'Isur. If so, the same applies to moving through a loaf or child.

Mishnah Berurah (308:26): The Shulchan Aruch is like the Rosh. Many Rishonim disagree, and say that a loaf or child permits only for a Mes, but not for other Muktzeh, even if the Muktzeh is a Kli. Many Acharonim rule like this. This is only if he puts in the Kli something not special for the Kli. If he puts in it something special for the Kli, e.g. garlic in a Meduchah used to dice garlic, to there is some Tavshil in a pot, all permit to move the Kli along with the Tavshil.

Kaf ha'Chayim (52): The Taz and Gra say that one should not be lenient like the Shulchan Aruch. Even so, to avoid a big loss one may rely on the lenient opinion. However, if one puts in it something special for the Kli, all permit moving it even from the sun to the shade.