I use 10-frame medium Langstroth boxes, and I find it very easy to build my hives with 3 slots down the insides of the short ends of the boxes. These slots are just a bit wider than quarter inch ply, half as deep as the frame hanging shelf is wide, and run vertically from the frame hanging shelf down to the bottom of the box.
These slots are 17/64ths inch wide, and 3/16ths inch deep. They are spaced so that one hive partition made of 1/4" plywood can go in the middle, making a double 5-frame nuc box, or that two hive partitions can be placed such that there are 3 chambers of three frames each for open mating (the center slot is not used for the 3 x 3 arrangement).
An additional advantage is that a partition made with queen excluder wire can be used to isolate a queen onto 3 frames of comb to lay eggs for breeding, making them easy to find. The workers can pass through freely. I call this my 7/3 arrangement, and I make up top-bar type sticks for the bees to draw out so the queens have new comb to lay eggs in.
Mating boxes have been made and used successfully with much less volume than 3 medium frames, but not without problems. The small colonies tend to swarm or abscond. To some, the use of bee resources in open mating is important to raise as many queens as you can in a short time, so they use the mini- or baby mating nucs, which go 30 to a super for drawing out the wax, and 3 frames and a mini-feeder are used in a mini-mating nuc. This allows a super crowded with worker bees to care for 10 queens for about 2 to 3 weeks during mating.
I don't use the mini mating boxes & frames. All my boxes are made with these slots, which are easiest to dado-cut before assembly, but can be routed with a small trimming router. I can use any box for 3 x 3-frame open mating, for 2 x 5-frame nuc increase colonies, for a breeder queen to lay in, or for a full 10-frame box. All my frames will soon be all mediums, so any frame for any purpose.
At present, my VQ's are open-mated 3 to a box with 3 frames each and two partitions. I prefer to put a lot of bees in the mating colonies, this encourages the newly-mated queen to lay like crazy, and her colony will shrink a bit for 3 weeks, then take off and increase on the fourth week, when I move them into the 2 x 5-frame double nuc arrangement. I'm in California where I don't need double screen boards for my nucs very much, but you may elect to make 3 x 3 and 2 x 5 double-screen boards if cold weather is a worry, mating your VQ's over a strong colony so they can benefit from it's heat.
My floor boards are flipable - one side is a screeened bottom board, the other side is either a double 5-frame nuc or a triple 3-frame mating nuc, with 1/4" square cross-section wood strips providing a slot for the bottom of the partition.
I tried to extend the slots down into the rims of the floors, but alignment from hive-to-hive didn't quite work out, so I just narrowed the partitions down in the area of the floors. There are no floor exits for the 3-frame and 5-frame arrangements - I just use cork holes (six to a box!) and I keep a bucket of corks handy.
I have 3 types of inner covers...3-frame mating nuc covers, 5-frame double nuc covers, and 10-frame inner covers. It is some saw work, but a very flexible system.
The partitions are made of 1/4" plywood and have "ears" that block the area where the frame hangar tabs go on the shelves, and they go all the way to the floor and extend up to just touch the top of the inner cover. Each chamber is sealed from the others, and I can add a piece of burlap, if necessary, to keep bees from fighting while the hives are open, but it is rarely necessary.
One last advantage to the slotted 10-frame hives over mini-mating nucs is that I don't have to wait for the bees to draw out the wax in the mini-frames, setting my queen rearing efforts about 10 days earlier in the year, or more, depending on weather.