I have adapted BMX and kids bikes into Mini velos.
Heres my Townend Megaforce 1 F1 BMX that I have.
F1 BMX was a short lived race series, for geared BMXs. A few manufacturers built bikes for it. Ive seen photos of a bike with full motorbike style fairing.
Try to get a bike that has a 25.4mm diameter seat pin. So that the seat tube has an outside diameter of 28.6mm. So you can easily mount a front mech. I tried to fit a front mech to a bike that had a thinner diameter seat tube, using home made shims. Never worked.
Front mechs are rated for a certain angle of the seat tube to chainstays. Bmxs with high BBs often have the chainstays pointing down to the wheel axle. where as bigger wheeled bikes have the bb lower than the wheel axle. It might cause problems.
You can fit internally geared hubs to BMXs quite easily. Shopper/ folding bikes often have the same wheel size, and the hub width is similar. Well for the old/low end 3 speed bikes.
Modern...er new school BMX, well the better models, might have a cassette hub. some have a really small cog. 9 tooth. And are to be used with a front cog of around 25 tooth. So giving a near 3 times gear ratio. Over 50 gear inches. I don't think theres any reason they cant be used with bigger front cogs, to get higher gear ratios. Might have problems using them with a tensioner. The hubs will likely have a thick, 14mm diameter spindle, instead of the older 9 to 10mm...er 3/8" diameter. So would need a suitable frame to fit, probably a modern one, upto about 10 years old. Older frames might be able to be filed, ground out to fit a thicker axle in the drop outs.
Ive had the idea of fitting road or MTB chainrings to a BMX crank. Ive not seen it done. Maybe theres problems. I thought one of those solid chainrings could be modded, cut off the teeth so that the diameter is less than the new chainrings. Then drill the holes in it for the chainring bolts. So would get round the cost of changing the whole cranks.
If you do want to change the cranks. You can get adaptor axles. That have all the bearings and cups that the BMX axle had. They have a square taper on them, to fit 3 piece crank arms. There ok. Im gonna fit that to my Townsend. But a lot of crank manufacturers are making cranks that don't have square tapers. You can get adaptor cups. That push into the BB shell. And have the standard English BB threading in them. I had a pair of cups from DMR, never got round to using them tho
If your wanting to fit a chain tensioner, or a rear mech to tension the chain. You can get a claw, that bolts onto the axle. Often seen on cheap bikes that don't have the gear hanger as part of the drop out. Ive managed to fit a claw backwards, so it more easily fitted on a rear facing drop out.
Modern multispeed bikes often have a removable gear hanger. So if you bash the mech it doesn't damage the frame. Theres seemingly hundreds of these. some might suit being bolted to a BMX drop out.