MITZVAH TANZ HISTORY
Shulchan Aruch Even Haezer 21- 5
It is forbidden to be served by a woman at all, whether she is an adult or a minor, whether a slave or free, lest he come to have forbidden thoughts. What service were they speaking of? Washing his face, hands, and feet,--
Rema. Some say that all this is forbidden only when they are alone, but in a place like a bathhouse where many people are found, one may be washed by an idolatrous slave woman, and this is the custom. And some say that anything not done in an affectionate manner where his intent is only for Heaven's sake, is permitted. Thus the custom is to be lenient in these things.
Bach Quotes the Maharshal “ For this reason(not done in an affectionate manner and in a public place) the Minhag is to dance with the Kallah (hand in hand*) so that her husband will love her or for the Kovod of her father, however, Talmidei Chachamim, should refrain” (dancing with the Kallah)
The Bach adds that in his country (Galicia) “the Minhag was, even for Gedolim to dance with the Kallah. Where the Minhag is to dance you may keep the Minhag, where there is no such Minhag then there is an Issur (to dance with the Kallah).”
Poskim who opposed this Minhag, (hand in hand dancing) called this a Mitzvah Haboh Ba'aveiroh.(=Aveiroh dance)
Eventually the Rabonim managed to abolish this Minhag of dancing hand in hand with the Kallah and instituted instead to dance with the Kallah by holding on to a napkin that was held on the other end by the Kallah.
Hence this new Minhag became a Mitzvah Tanz** instead of an Aveiroh Tanz.
(Chasidim replaced the napkin for a Gartel a) had no napkins- b) longer and further away from Kallah)
For the father and grandfathers of the Kallah there never was a problem holding hands so the original Minhag remained and they dance holding hands.