Okay, completed my test. It's a rough test, but told me what I needed to know.
the TL

R version is Colonial offices (didn't test taxes since it's much further up the civics tree) and Casa De Contracion do work if you move the capital before putting them in. Actually Colonial offices will work either way, but if you have Colonial offices slotted when your move capital project completes it won't work. You will have to remove the card, then put the card back in (I did it the subsequent turn). Casa de Contracion I could not get to work if I built it before moving my capital, but it worked just fine if I moved capital before this finished (it's okay if it's in progress).
I didn't record my growth numbers for the move capital first test, but rest assured the numbers did go down like they should; here are my production numbers turn before and turn after Casa De Contracion finished (both cities have governors, my capital didn't have a governor at the moment, so I didn't include that one). Obviously both of these cities are on my original continent.
production turn before/turn after completing Casa De Contracion
Sidon: 17.8 / 20.4 seems to be slightly under 15%
Biruta: 25.2 / 28.8 slightly under 15% as well
Now if I allowed Casa De Contracion to finish before completing the move capital project and I have colonial offices already slotted here are some numbers. sorry for poor formatting. All 3 cities had governors (so I could check for both things).
production (turn to grow in parenthesis) turn before move capital completes/turn cap project complete / turn after cap project complete/
Sidon: 18.9 (34) / 18.9 (33) - no change / 18.9 (32) - no change
Biruta: 19.9 (39) / 19.9 (38) - no change / 19.9 (38) - no change
Tyre: 24.1 (9) / 19 (8) - prod loss due to no capital/ 19.9 (7) - small increase from something
My last thing I did was spend money to unlock cards, take colonial offices out and then put back in the next turn.
Sidon 18.9 (31) / 18.9 (28) - production same but growth did go up
Biruta 19.9 (36) / 19.9 (32) - production same but growth did go up
Tyre 19.9 (6) / 19.9 (5) - no real change, but growth was close to pop already
I played a few more turns after that to see if Casa De Contracion bonuses would trigger, but they didn't. Not sure if there's any way to get them to trigger. Maybe move capital again to a 3rd continent. Seems unlikely though.
I'll include a save if anyone wants to mess around with it. This is 1 turn before the capital project completes. For the 2nd half of the test I switched production, for the first part I just let it complete.