I've done some exercises for folks during some classes to help people with the ukemi for koshinage that involves one person on "all fours" and the other person draping then "threading" their bodies over then under their partner. It's mostly an exercise in tucking back into nage's body.
As far as where the hand goes, it depends on the koshinage. Sometimes, you can grab onto nage's arm if they offer it. Other times, you're just going to have to mess up their dogi by grabbing their lapel or even their belt. And, heck, other times, you're just going to have to fall without the use of the hands, especially in certain koshinage where nage holds onto both hands (eg jujinage koshinage) so uke is both unable to grab nor unable to slap. Nasty, those.
As for "lifting with the hips," I'll say that the best koshinage that I've felt pretty much had no "lifting" whatsoever. Rather, nage manipulated my balance in such a way that their hip went underneath my center without having to lift. It's kind of like the best footsweeps I've felt (from an 8th dan judo practitioner) -- just literally a tap on my ankle and, because he'd manipulated my balance exquisitely well, there was no way I could do anything to save my face from getting planted without taking a breakfall...