Pretty simple to do, but it does take time (3 to 4 weeks) to cycle brood and draw new combs.
There are a few pre-requisites that will help the transition go off smoothly. First, you cannot do this without queen excluders. Well, you could but it will go so much easier and trouble free with them. Get some steel ones. Do not use plastic QEs. The hive also has to bee of sufficient size to populate the boxes during the exchange and to be able to build-out and organize themselves into the new boxes without interruption. So, this is best done if the hive population is fully occupying a minimum of 2 deeps in size. Am also going to assume you will be drawing a lot of new medium combs. So this also needs to be pre-planned and done when there is a steady heavy nectar flow on. Or else you have to be prepared and setup to do alot of feeding of syrup to draw new combs. Avoid the feeding scenario, as that seriously complicates what to do with the deeps at the end of the process. So, the planning and preparation criteria that must be met are: steel queen excluders on hand, hive(s) population is occupying a minimum of 3 deeps, a strong nectar flow period of 3 weeks.
Once those criteria are met, the process involves getting the brood nest started in a medium and waiting for the nest in the deeps to time out while moving the deeps up in the stack. Along the way, manipulating the hive so the bees are drawing combs in a manner that the bees can handle the extra work as seamlessly as possible.
Step 1: Take the hive apart down to the bottom board. Place an empty deep box on the bottom board (no frames in the box). Take all of the brood boxes apart. Find any and every frame that has eggs and/or brood on it, place those frames into that empty box on the board. While doing this, find the queen and move her into that bottom box. Consolidate all the brood in the bottom deep box. Keep the order of the brood frames, which is next to which, the same as you do this. If there is more brood that needs to go into a second deep that is fine. Just get it all as low in the stack as possible. Place a new medium box directly on top of the deep brood box(es). Place a queen excluder on top of the medium and stack above the excluder whatever other boxes there are to the hive, as needed for hive size and flow conditions.
Step 2: One week later. The queen should be roaming in the medium, spreading her pheromones in it, the bees will have drawn some of the combs, and the queen may have layed a few eggs. There may even be bit of developing larvae. Again, take the hive apart down to the bottom board and reverse the boxes. Place that starter medium on the bottom board. Find the queen and make sure she is in that bottom medium box. Place a queen excluder directly on the medium. Place all remaining boxes above the excluder with brood boxes being the lowest in the stack. Yes, this confines her to just the medium on the bottom. It seems a small space, but there is a reason for doing this, and it is short term.
Step 3: More than 6 days but absolutely no more than 8 days from the reversal date. Take the hive apart down to the medium box. Remove the queen excluder and check the bottom medium for queen and a stable brood nest being established. Place 2 new mediums on, making the bottom of the hive 3 mediums which the queen has full roaming access to. Put the queen excluder on top of that third medium. Place an empty deep box (no frames at all) on the excluder. Take each frame out of the deeps one by one and shake all the bees off. You may shake the bees into the empty box on the hive or directly out front of the hive entrance, your choice. THOROUGHLY inspect every bee-less deep frame for queen cells and destroy all of them. Put each bee-less deep frame into the empty box on the hive one by one as the qc seek and destroy check is completed. Repeat for all deep boxes and frames, while consolidating all brood frames into that first deep in the stack. Everything else goes above. Close up the hive.
Step 4: One week later. Check hive size and honey space. If they need more space, you must under super. Meaning move the deep(s) up in the stack and place new medium(s) under the deeps, and directly above the queen excluder.
Step 5: One week later. Any and all brood that was in the deeps has emerged. The deeps are now highest in the stack and are being used by the bees as honey supers. Harvest the deeps, exchanging them for mediums. Extract and bottle the honey. Some pollen is expected to be left in the deeps. This is ok and may be salvageable, depending on what your end plan for the deeps is.
Step 6: At this point you have extracted empty deeps of drawn comb (that may still have a bit of pollen in them). High value equipment! Drawn comb is worth more than gold, to the beekeeper. Take the frames out of the boxes. Setup your table saw and cut the bottoms of the frames off to get them down to medium size and staple or nail in new bottom bars. Reset the tablesaw, cut the bottoms off of the deep boxes to make them medium box size. Put the cut frames into the cut boxes et voila, you have your repurposed/recycled boxes of drawn comb (med brood chambers) ready for splits etc.
What has happened? In step 1, the nest was started (primed) in a new medium sized box. In step 2, the new box had enough combs started that the queen could continue laying without interruption. She was moved down and confined to concentrate the establishment of the nest in the bottom of the hive. In step 3 the brood nest space was largely expanded, returning to a normal bee-comfort size of 3 mediums (or 2 deeps). When brood is separated from the queen and placed above an excluder, there is high probability that the youngest bees tending that brood above will lose track of the queen and will start cells. Step 3 shuffling of frames and checks takes care of those to ensure hive stability. Steps 4 and 5 are essentially back to normal business of managing the hive while waiting for the old nest brood to mature and emerge out of the deeps. Once the brood in the deeps has cycled, the deeps can be removed and either repurposed, recycled, or disposed.
By pre-planning to satisfy the criteria and following those steps on schedule; the bees` transition into the mediums will be smooth, seamless, and trouble free. Miss a criteria, miss a step, or miss in timing and you will have some problems.
There may be other thoughts and other methods. The above is what has worked well for seamlessly moving a colony into different equipment when there are brood and resources (honey/pollen). When there is little to no brood present, you could just shake the colony jnto the new equipment and feed feed feed.
Hope that helps!