Is it obligatory to move to Madinah next year based on the vow, or can it be postponed for four years until the children finish their studies, and is dwelling there for a month or two considered a move that fulfills the vow?

If the vow to reside in Medina is an act of devotion to Allah, then it must be fulfilled immediately if a time was specified for it; otherwise, according to the majority of scholars, it may be fulfilled at leisure. It is recommended to reside in the vicinity of Mecca and Medina, unless there is a strong presumption of falling into blameworthy acts. If you are unable to fulfill the vow, then you must offer an expiation for breaking an oath.