I would add that you need to see where they source the parts. For example, we source from Digikey/Mouser/Etc and then use a local assembler so that we can guarantee the provenance of the parts. This means no substitutions by the assembler with grey market parts. We can also deal with them easily and in the same language. This also means we can pick them up and packaging and damage during transit is less of a problem.
Will you be supplying a test rig? I believe the further away the assembly house the more it is important for them to pick up mistakes so they can fix them. This also reduces shipping costs and time spent sending things back for repairs.
You can also look at partial construction, ie jelly bean components where you don't mind substitutions, and then assemble the rest at your facilities.
I would say before sending it off, try reading up on some Design for Manufacture theory so that you can reduce your manufacturing steps and associated costs. It's all about the cost margin
Edit: Also I should mention the process will change depending on what version it is, and how many you want to produce. Making everything surface mount can be a pain to solder by hand, but really streamlines things if you use a mass production service. TH is the devil!