Is it obligatory to fulfill a vow to carry the bride from her family's home to her marital home after marriage? And can she be carried in a car, or is expiation due for this vow?

This type of vow falls under the category of permissible vows, and the majority of scholars hold that it does not necessitate any obligation. As for the Hanbalis, they give the one who made the vow the choice between fulfilling it or offering an expiation for a broken oath (kaffarat yamin). Since carrying the wife in the intended manner is impossible, the questioner must offer an expiation for a broken oath.