This is the only type split I do, this may be just me but seems the best results I have found when letting them raise their own queen has been when the small flows are going before the main flow in spring (tulip poplar) and the hives are building very fast (and have plenty drones) is to take the queen on the frame she's on and 1 frame capped brood, 1 frame of larva, 1 frame of honey and 1 frame of foundation and put in a nuc (queen frame and brood frames with attached bee's) , just make sure you leave a frame of eggs in the main hive, let the main hive with all those bee's raise a queen, oh and I guest its because not many young bee's but it seems a queen less hive will gather more honey.
And the laying queen will build up that nuc into a hive in no time. the main large hive with a flow going will feed the larva plenty, if you end up with a couple frames with eggs on them and you have a few mores hive, take the extra frames with eggs on them and steal a few frames of bee's from the other hives and make some more nuc's.
by taking queen and stealing frames of brood with bee's attached is like imitating a swarm, some times it needs to be done more than once but it does keep them in check, I dont have much swarming because I take frames from my hives most of the year to start nuc's.. some beeks might do other ways but this has worked for me and made some fine queens.
I do graft and raise queens and like Robo said you do have a better chance getting a better queen if you graft because you can control things like feeding and timing, but when a flow is going the bee's themselves can raise some great queens also, I dont care to split hives when there isn't a flow. but this way swarming hasn't been a problem at all the last few years.
Now I don't split all my hives but do split a few, taking brood frames from hives to make nuc's for grafted queens keeps swarming down also.