You will need a USDA stamped health certificate for the EU (filled out by your vet), as USDA stamped airline health form (filled out by your vet), and an import permit from the Swiss, and a CITES export permit from the US.

NB. The import permit from the Swiss is only valid for 3 months. It can take at least that just to get a CITES permit. However, ask the BVET for an import permit and tell them what it's for and that, when you are ready to import and have all your other docs, you will need to have your permit re-issued.

Also, you WILL need a CITES permit. We argued long and hard with the US FWS about appendix II animals. We had a couple of knowledgeable people argue with them that their interpretation is incorrect but to no avail (big surprise). It will take at least 3 months to get from the time you submit your paperwork.

Also note that the airlines will not transport animals if the temperature on the tarmac is too hot or too cold. Thus, if you got your CITES permit issued to you in a few months, it will be the winter and you probably couldn't get your animals to be transported on an airline until the spring. I think the tarmac temp can't be below 45 F or something.

I haven't been successful at this yet, we are just getting our docs together for the CITES permit now (note that you will need chain of ownership papers proving things like the animals were bred domestically and not imported, etc). We are trying to import two male eclecutus parrots.