I guess they should work together, the are the same type of bike basically from what I can tell. The BB shells are probably the same 68 mm.
What is going on with the Moby seattube? It looks like an elliptical tube but with a round seatpost?
The bottom bracket and crank will transfer over just fine. You'll need a bottom bracket tool and a crank puller for ISO square taper. Also a lockring tool, some good lithium or molybdenum grease like Phil Woods (or go to Autozone). Allen wrenches to swap over the seatpost, saddle, stem, handlebars. Metric, probably a 6mm, 8mm, 10mm. You probably already have a 15mm wrench for removing the wheels. And you'll need a chain breaker. Get a good one, the cheap ones are a pain to use.
You really think it's a downgrade? I got a deal on the Bowery, so I bought it even though it's a bit small for me. Now I want a larger frame. The slanted TT on the Bowery also erks me. I want a track style frame and the BD seemed as good as any, but if you can recommend a better one that doesn't cost an arm and a leg I'd definitely consider it.