If I move to France can I use U. S. Series 4 with new French carrier, such as Orange?

A watch purchased in USA will be incompatible with European cellular networks, and the cellular functions will not work. See https://www.apple.com/watch/cellular/

Yes. The watch will have all the functions it normally has when you are back in USA (subject to a few things being not available in France). By default the watch uses the pairing link over Bluetooth to pass all its traffic via your iPhone. That Bluetooth will continue to work with your watch exactly like when you were in USA.

The watch will not be able to connect standalone to cellular networks outside its designed region (North America). This will not reduce the functions available because watch cellular only goes active when the iPhone route is unavailable, usually because the phone is powered down or you walked away out of Bluetooth range.

I mentioned above some limitations on function availability. That limitation is described in watchOS - Feature Availability - Apple and applies to all devices whether visiting with a roaming phone, or purchased in the subject country. Quite simply, Apple does not support all functions in all countries, and USA seems to get unfair priority.

See my last reply. I’m not planning on researching the precise technical reasons, but I can assure you Apple Watch does not migrate between different cellular regions. Unlike iPhone there is not enough space in the watch casing to make it happen.

Any Apple Watch will pair over Bluetooth with any recent iPhone. The Bluetooth function is common to all iPhones and completely independent from the cellular side. Bluetooth doesn’t care which network or SIM you are using, it doesn‘t know if the phone is roaming or on the home network. It doesn’t even care if the phone is off cellular network, Bluetooth will still connect between watch and iPhone.

I can’t give you a technical reason without several hours of research into cellular bands and operational methods. It is sufficient to say

1. Apple didn’t take on the cost of designing and manufacturing multiple hardware versions without good cause.

2. There is more than enough evidence in posts here on ASC, from users who tried to take phones sourced from USA and use them on networks elsewhere. They failed.

You have a choice, wait... or go with the GPS-only version which will always work through your paired iPhone, anywhere in the world where the iPhone has data service. If you don't need the standalone cellular function that is an acceptable solution. In fact even the cellular models make most of their communication through the iPhone, and cellular is only used when the iPhone is not reachable. The cellular radios are actually off until the iPhone is lost, then cellular comes to life and looks for a connection.