On Yom Tov may one carry his Daled Minim back home after having used it for the Mitzvah?
[It is forbidden for one to carry on Yom Tov in an area without an Eiruv for a non-Yom Tov need.] it is forbidden to carry the Daled Minim in a Reshus Harabim [i.e. public domain] for no Halachic need of that day. [This prohibition includes even carrying the Daled Minim through a Karmalis that does not contain an Eiruv. It is however permitted to carry it in any city that contains an Eiruv that permits carrying on Shabbos, even if one is carrying it for no need at all. Likewise one may carry it in a courtyard from one house to another, or from one courtyard to another even if Eiruv Chatzeiros was not performed.]
May one carry a set of Daled Minim on behalf of women? It is permitted to carry the Daled Minim through a public domain for the sake of women shaking Lulav.
May one carry the Daled Minim back home after concluding the Mitzvah? If one originally carried the Daled Minim to Shul on Yom Tov, then if he suspects the Shofar may get lost or stolen if he leaves it in Shul, he may return it home on Yom Tov even if he has no more need to shake it that day. If however one brought the Daled Minim to Shul from before Yom Tov then he may not carry it back home on Yom Tov unless he needs to use it there for the Mitzvah. [This applies even if one suspects the Daled Minim may get stolen if one leaves it in Shul. If however one has a safe area to leave the Daled Minim in Shul, such as a locker, then he may not carry it back home even if he brought it there on Yom Tov, unless he plans to use it at home. If one carried the Daled Minim from Shul for the sake of Mivtzaim he may return it back home even if it was brought to Shul before Yom Tov.]
It is forbidden to carry the Daled Minim on Yom Tov for no need in any area that does not have an Eiruv and is hence forbidden to carry on Shabbos. If however one originally carried the Daled Minim to Shul on Yom Tov, then if he suspects the Daled Minim may get lost or stolen if he leaves it in Shul, he may return it home on Yom Tov even if he has no more need to shake it that day.
*Important note: The laws below are only relevant to an area that does not have an Eiruv and hence carrying on Shabbos is forbidden, in which case even on Yom Tov carrying is only permitted for a need.
May one carry the Daled Minim to a Shul on the 1st day on behalf of shaking it on the 2nd day?
May one carry the Daled Minim back home after bringing it to a friend’s house?
If one does not plan on using the Daled Minim anymore that day then it is forbidden to carry it home if the Daled Minim has a safe resting place in the home he is currently in.
Returning home from a meal: Based on the above, if one is eating the Yom Tov day meal by someone’s home he may not carry the Daled Minim with him after finishing the meal for the sake of bringing it back to his house unless he plans on using it or suspects it may get lost or stolen.
May one carry the Daled Minim on behalf of a person of which it is questionable if he already fulfilled his obligation?
May one carry the Daled Minim on behalf of a child?
 See regarding Shofar: 596/2; M”A 596/2 in name of Raaven; Elya Raba 596/1; P”M 596 A”A 2; Chayeh Adam 142/19; M”B 596/4
 See Admur 518/1-4
The custom of the Rabbeim: To note that seemingly due to this reason the Rebbe Rashab and Rebbe Rayatz were particular to never carry on Yom Tov. [See Otzer Minhagei Chabad 359 that the Rebbe Rashab did not carry on R”H and by Tashlich he would give his Siddur and glasses to one of the Bochurim to wear. Likewise in Reshimos Devarim p. 180 it states that the Rebbe Rayatz once saw a member of Anash carrying his Tallis on Yom Tov and he chastised him for doing so. I also heard from Harav Eli Landau, R.Y. of T.T. Kfar Chabad, that his father, Harav Yaakov Landau, would walk to the Mikveh together with the Rebbe Rashab on Yom Tov and the Rebbe Rashab would ask him to carry his towel for him to the Mikveh. Thus the Rebbe Rashab and Rebbe Rayatz were particular not to carry. However the Rebbe was accustomed to carry on Yom Tov as was witnessed on various occasions [i.e. removing keys to open door etc].
 The reason: As by doing so one is carrying the Daled Minim from one domain to another without any Yom Tov need. [Admur ibid]
 M”E 596/2; Kaf Hachaim 518/14; 596/6; Piskeiy Teshuvos 589/5
Ruling of Admur: See Admur 518/1-4 that the prohibition applies to even a Karmalis and it is only permitted if there is an Eiruv. However from the wording of Admur here “not to carry in a Reshus Harabim” it implies that it only applies to an actual Reshus Harabim and not a Karmalis, and so seems to learn Kaf Hachaim 596/6 in Admur. Vetzaruch Iyun.
 Admur 518/2
 Admur 518/3; Rashal brought in Taz 518/2; See Kaf Hachaim 518/14; M”B 518/10
Other Opinions: There are Poskim that forbid carrying an item for no need at all even though a courtyard, unless Eiruv Chatzeiros was performed and it is hence permitted to carry even on Shabbos. [M”E 596/2 [regarding Shofar]; Taz ibid; Piskeiy Teshuvos 589/5; See Kaf Hachaim ibid for list of opinions on this matter.
 See regarding Shofar: Admur 589/2; Ravaya; Rosh; Tur; Beis Yosef; Chida in Yosef Ometz 82; Kaf Hachaim 589/25; Igros Moshe 3/94; Az Nidbaru 13/38 [permitted even for female children]; Ketzei Hamateh 589/5 [“so is the custom”]; Piskeiy Teshuvos 518/4; 589/5
Other Opinions: Some Poskim rule one may not carry the Daled Minim through a public domain that does not contain an Eiruv simply for the sake of women. [Shaagas Aryeh 106 brought in Machazik Bracha 589/5; Shaareiy Teshuvah 589] The Chida in Yosef Ometz ibid negates this opinion and concludes like above. See Piskeiy Teshuvos ibid footnote 16 that perhaps today it is permitted according to all opinions as women have accepted upon themselves to hear Shofar and it is hence like a vow which is Biblical and one may hence carry on their behalf according to all.]
 518/1; Rashal; Taz 518/1; M”A 518/1
 The reason: As the same way the Sages allowed carrying the Shofar for the sake of the Mitzvah similarly they allowed returning it back to prevent it from getting stolen or lost, as if they were to prohibit returning it one would refrain from carrying it out in the first place. [Admur ibid; Taz ibid; M”A ibid]
 The reason: [As it is forbidden to carry for no need on Yom Tov and it was only permitted to carry it back home when brought on Yom Tov cause one may refrain from bringing it otherwise]. However by forbidding returning it on Yom Tov will not prevent people from bringing it before Yom Tov as they can always also return it before Yom Tov. [Admur ibid]
 Implication of Admur ibid; Taz ibid; Rashal ibid and so writes M”B 518/7 in their opinion that one may never return it simply to refrain from monetary loss.
 Piskeiy Teshuvos 518/9; Mishneh Sachir 2/138
 As the bringing of the Shofar to Mivtzaim is considered carrying it on Yom Tov for a purpose, of which one is allowed to return it on Yom Tov even to merely save it from getting stolen. Upashut!
 Piskeiy Teshuvos 589 footnote 7; Upashut
 Piskeiy Teshuvos 518/9; See Teshuvos Vehanhagos 1/348; Shraga Hameir 5/72
 Shaagas Aryeh 107, brought in Machazik Bracha 589/6; Shaareiy Teshuvah 589; Kaf Hachaim 596/29; This certainly applies according to Admur 589/2 that rules it is permitted even for the sake of women
Other opinions: The Machazik Bracha ibid [brought in Shaareiy Teshuvah ibid] questions whether this is allowed as according to the Rambam the need to be stringent by a Biblical doubt is only Rabbinical and hence how can one allow carrying in a public domain on Yom Tov in a case of doubt. The Kaf Hachaim ibid concludes that it may be done as according to most Poskim all public areas today are the status of a Karmalis. According to Admur ibid this is not necessary as it is permitted to carry even on behalf of blowing the Shofar for women. Thus we see he holds there is never a prohibition involved in carrying for a need even if it is a mere custom.
 See regarding Shofar: Admur 589/2 [regarding women who are completely exempt from the Mitzvah and certainly it would apply for children]; Igros Moshe 3/94; Az Nidbaru 13/38 [permitted even for female children]; Ketzei Hamateh 589/5 [“so is the custom”]; Piskeiy Teshuvos 518/4; 589/5
Other Opinions: Some Poskim rule it is forbidden to carry a Shofar for the sake of a child, or for the sake of the blows in Musaf. Likewise they rule one may not carry a Lulav for the sake of shaking it in Halel. [Shaagas Aryeh 108; Machazik Bracha 589/7; Shaareiy Teshuvah 589/1; Kaf Hachaim 589/30]
 As Admur 589/2 permits even regarding women who are completely exempt from the Mitzvah thus certainly it would be allowed for children that have reached the age of Chinuch and are hence Rabbinically obligated.
 Az Nidbaru 13/38