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Mishnah (Beitzah 1:5) records that Beit Shamai do not permit carrying a child, lulav, or Sefer Torah in a public domain on Yom Tov, while Beit Hillel do permit such carrying. Our Daf(Beitzah 12a) explains that Beit Hillel believes that “mitoch shehutrah hotza’ah letzorech. hutrah nami shelo letzorech”, “Given that the Torah permits carrying for the sake of food preparation, it permits carrying for any Yom Tov need,” while Beit Shamai rejects this expansion.
This concept is commonly referred to as “mitoch” and applies to all melachot that are permitted on Yom Tov. The Halachah follows the opinion of Beit Hillel (Shulchan Aruch O.C. 518:1).
Even Bet Hillel would agree that there needs to be some purpose in carrying in order for it to be permitted on Yom Tov; lugging around rocks is forbidden even according to Bet Hillel. The purpose can be the needs of a mitzvah – like carrying a lulav to the synagogue or a sefer Torah to study from, or the needs of simchat Yom Tov, enhancing the joyousness of the holiday.
The Torah states No work is to be done on those days, except to prepare food for everyone to eat - that is all you may do (Shemot 12:16), explicitly permitting work that is done for the preparation of food. We question the scope of this leniency: Which labors are permitted for the sake of okhel nefesh? Mi-Tokh She-Hutra Le-Tzorekh