That's a truly difficult question to answer. If you enjoy those particular guns, and like them, then shooting them would probably have its own intrinsic enjoyment beyond anything special about the chambering being fired.
.38 S&W (not special) is most commonly encountered with a 146gr LRN bullet. It is offered in this loading from both Winchester and Remington. I like the Winchester version somewhat simply because it still comes in nickel cases. Ten-X and Old West Scrounger also offer it in variant loadings, and I think Magtech might make it as well. Factory ammo isn't cheap and the brass is, for some reason, hard to get as late, or so I'm told. A forum friend was helping me scrounge some for my topbreak. Used to be that folks that loaded a lot of .38 Special would sometimes just give you .38SW brass that got mixed in or toss it, those days seem to be over.
It'll basically be a light recoiling offering. From a K frame, I'd think that it would kick about like wadcutters since velocity is mild. Handloads for a relatively modern K frame like a Victory model could probably be made a bit more potent than current factory offerings. I've only shot it from topbreaks so I'm extrapolating about the K frame experience.
There won't really be anything novel from a K frame so chambered - in other words it isn't going to do anything that a similar revolver in .38 special wouldn't.