I make my British National brood boxes from flat-pack kits (seconds) from Mann-Lake. And by 'seconds', they really do mean seconds ... (to be pedantic, they call these 'Standard' Grade ... ) But - to be fair, these DO have a low price tag.
This design of box has upper and lower external rails, and where the inner planks make contact (or don't) with these rails, there are gaps. And there are also gaps in the corners. Gaps, gaps gaps ...
So - after gluing-up these boxes with 100% waterproof glue, I then run a bead of the same glue down into each gap. If the gap is really serious (> 1mm) - and some are - then I knife polyester automotive body-filler into those large gaps. I also run some glue over any knots, so that they're fixed for life.
Then I paint the boxes - inside and out. Why ? Because the outside is exposed to the weather, and the inside is often as wet as a shower room with water running down the walls, as the bees evaporate moisture for their living.
And - contrary to the common mantra - bees will NOT propolise the interior surfaces of a box during their first year of occupation, and so that is when the wood is most at risk from swelling and busting glued joints as a consequence.
BTW - I use a product called 'D4' glue, but you won't have it. Titebond or similar would do the same job. You could also use polyester automotive body filler (cheap enough if you buy in 5 litre drums), with extra polyester resin mixed in to make it runny.
Good question, by the way - I've not heard that one asked before now ...
LJ