PCB design and manufacturing: How important is it to select actual vs. generic parts?
You can ship your own parts to them.

You may also specify parts where the assembler supplies their own parts for really generic components like caps and resistors. They should have a bunch of these just on-hand.

Other times you specify parts and they order the parts themselves in a key turn solution.

But really, these are all very vendor specific issues that you NEED to discuss with them. Certain circuits are pickier with certain "generic" components than others. Surely, high frequency or RF boards can't just take any old SMD capacitor or resistor of the same value that will fit onto the pad.