I sure wouldn't and never have. I take my frames of brood for the nuc and shake another brood frame of bees into the split and add the queen and leave them in the yard. I want the old bees who tend to be hard on a new queen to go home to the old hive leaving young receptive bees to release and love the new queen. Make sure you don't get the queen when selecting those brood frames. A good fast way to do that is to brush or gently shake the bees off the frames for the split and put them in a different box. Put a queen excluder over your top brood box of your original colony and set the beeless brood combs on top. Leave for a half a day or overnight and the frames will be coverered with nurse bees and be guaranteed queenless. Set that box on a new bottom and add your caged queen. Check in three days to see if she has been released. If bees are all attending and not attacking the cage just release her and stay out for a couple weeks as bees can ball a newly released queen when disturbed. I have seen it happen. Good luck.
I've done a few splits (very few) with mated queens and have not blocked the entrance, all have been successful.
If you are trying to keep the foragers in the nuc I've added a robber screen and then pushed leafed twigs in the RS. Don't stuff it full, just enough to make it a little difficult for the bees to leave. I've seen them orientate to the new nuc this way, some return to the nuc some to the mother hive.
We don't block the entrance/don't try to keep the foragers in the split. Give them a good mix of brood ages, lots of young bees, wait a day so that they realize they are queenless then introduce the new queen in her cage. We normally see bees heading out to forage within a few days of doing splits.
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