Many people seem to believe that bees won't swarm if you have lots of supers on. I have not found this to be true. It HELPS to have room, but the aim of a hive coming into spring is to throw off a swarm.
The squence of events is:
Start rearing brood.
Build up the population by having a "mature" brood nest filled with brood.
Backfill the brood nest to shutdown the old queen (to prepare her for flying and free up the nurse bees to swarm).
Start swarm cells.
Cast a swarm composed of the young nurse bees and the old queen.
Some people interfere with this sequence by swapping the bottom boxes and splitting up or disrupting the brood nest. This DOES prevent swarms. Unfortunatly it also requires a lot of work on the part of the bees to rearrange everything.
I prefer to pull a couple of frames (not adjacent) of brood out and move them up to the next box. I don't do this unless the brood nest is full and there are a lot of bees. This gets the brood nest to grow up to the next box, baits the bees up and leaves gaps in the brood nest for the bees to fill. The young bees get busy drawing comb and growing the brood nest so the "mature" brood nest doesn't occur and they don't get to the backfilling the brood nest. If they ARE backfilling the brood nest with honey, I'd pull the honey out and put in empty frames to open it back up. If they are already rearing swarm queens, I just put each frame with some queens in a two frame nuc with a frame of honey and let them rear a queen. This relieves congestion, opens up the brood nest and gives the bees the impression there has been a swarm. If they have capped cells, I'd pull the old queen and half the brood out and do a split. BTW I love to put in actual empty frames. No foundation. They don't even need a comb guide or starter strip because the drawn comb on each side serves as a guide. Between two frames of brood they draw it very nicely in the frame, have a consistent thickness, AND it's usually very small cells.
What I prefer, for both swarm control, honey production, and increase, is to do a split two weeks before the flow. (or right at the start of the flow if you miss that). Take all the open brood and the old queen and put them in a new place and let the old field bees and all the capped brood (and at least a few eggs) stay at the old location. The old hive will rear a new queen. This causes them to store more honey, because they have no brood to feed, have a break in the brood cycle which helps with the Varroa, and you get a new young queen.
The old hive won't swarm because they are queenless. The new hive won't swarm because they have no field force.