I have had 1 hive swarm 4 times. caught all four. swarm 1 is doing fine, swarm two looks to be queenless, swarm 3 I gave away, and swarm 4 is doing great too.

after catching swarm 2 I gave them a frame of brood from the mother hive to help them get started, and 9 frames of plastic foundation in a deep box. that was on 6/1. ( 4 weeks ago). They have barely drawn any new comb, and basically just sit on the 1 frame with comb that I gave them to start with. I haven't seen the queen, and no new brood, though it's hard because they sit so tightly packed on that 1 frame you can barely see the comb. I am fairly certain this hive is queenless. (did not return from mating flight?)

Swarm 4 is a week younger (3 weeks now). I put it in a deep with only 5 frames, flanked by campaign signs on either side.
Within 10 days they had those 5 frames almost completely drawn. all but the 2 outermost sides. I did see the queen then, but no brood. today two frames had nice half circle patterns of capped and open brood on both sides.

My plan for swarm 4 had always been to recombine it with whichever was weakest. And the plan was to use the newspaper method. But that was when I was still expecting the other hive to also have a queen.
Now that I have 5 frames with a queen laying nicely, and basically 1 and 2 half frames with comb, honey, and pollen in it, and a couple 1000 bees on it, can I just add those 3 frames and bees into the deep that has the 5 with the laying queen? Or do they still need some form of introduction separation?