France, Croatia

Go to Bordeaux airport from Sarlat and fly to Croatia.
You can use Google Flights to see all the airlines that have service from France to Croatia by searching
From BOD To Croatia.

Where in Croatia did you want to go? Because of its "boomerang" shape, it can take some time to get from one part of the country to another. In other words, a flight to Zagreb won't get you anywhere near Dubrovnik.

In addition, flights to many of the airports in Croatia are only during "the season" (summer); depending on your exact travel dates in the fall, you will have more or fewer options for getting to places like Split and Dubrovnik (Zagreb has more year-round service).

Look at Skyscanner for flights to various cities from Bordeaux, for you exact dates (remember, in addition flights not operating year-round, many flights are only on certain days of the week): https://www.skyscanner.com/

Returning to the West Coast of the US, prepare for two connections, and potentially high prices. Even from New York, flights to and from Croatia are expensive, and often involve less than ideal connections.

Toulouse Airport is also an option out of France, coming from Sarlat. Larger airport. Volotea flies to Split until the end of October, twice weekly.

Thanks for the information. We have read the Bruno detective series, set in the Dordogne region on France, hence wanting to visit Sarlat. We plan to hug the coast in Croatia, Dubrovnik, Split and Pula. Any suggestions?

Croatia isn't that easily done by train. Pula, for example, doesn't provide rail service to the other places mentioned in Croatia.

You don't state how much time you have, but unless it's a lot, I'd focus on either Istria or Dalmatia in Croatia for that portion of your trip. You could do Novigrad/Rovinj/Pula (or reverse) in Istria by bus or car, flying in and out of Pula, or you could fly to Zadar, Split, or Dubrovnik and hit those locations by bus or car, or include a visit to an island by ferry.

Another option is if you find cheap flights into Venice from France, you could ferry to Croatia.

You'll find trains not really helpful for a trip along Croatia's coast. Google Croatia Rail Schematic to see where the rail lines are. You'll need to rent a car or use buses. The buses are comfortable, in my experience.

Or you can hop down the coast with a combination of ferries.

You need a minimum of two full weeks if you want to do Pula - Split - Dubrovnik (and some of the stuff near/between them). We've been to Croatia twice, total of almost a month to cover the area from Dubrovnik up to the Istrian peninsula. We did everything by public transportation except for a car for 4 days for the hilltowns in Istria and to get to the Plitvice Lakes. But both trips were in summer when ferries are more frequent. We took a ferry from Dubrovnik to Korcula (you could do Hvar same way) and another ferry to Split on the first trip. The second trip (just got back) we took buses in Istria to visit Pula and Porec from Rovinj, then we had the car for the hill towns to Plitvice to Zadar, then buses again to Sibinek to Split. Then ferries to Hvar and Brac.

You'd have to see how late those ferries run. I know they run year round but on a less frequent schedule. Buses run year round. If you aren't interested in hill towns you would not need a car at all.

As mentioned, flights (easy jet, Ryan air, etc.) are seasonal, usually into October though. But also not daily so you have to work out a schedule to have the correct number of days and fit it into the flight schedule. For Croatia we have always had our transatlantic flights in/out of Italy or France and done the budget airlines to/from Croatia. To fly to the US from Croatia is more expensive and requires changes anyway so we plan on at least some time at both ends of the trip in another country. You might find more frequent flights from Croatia to London if you don't mind spending a night or two there, then there are lots of flight options back to the US. But don't plan to fly from Croatia to London on one airline (e.g. easy jet) and then back to the US on the same day. Too risky if that first flight is delayed. Plan on at least one night (I always do at least two).

Check out rome2rio.com for all the possible ways to get between these places.

We want to see the Dordogne region, while in France. Any suggestions of villages to stay in, besides Sarlat?
BTW, we will rent a car in France and in Croatia.