Is there any restriction when moving money from France to the US via bank transfers? Do I have to declare anything? Can this trigger some tax event or audit, and if so is it any threshold above which those would be triggered? The transfer is done between a bank account in France belonging to me and a bank account in the US belonging to me.
There is no limit, and there are no tax consequences.
However, if you own money in a french account, you were supposed to report that in FBAR filings (foreign accounts) in your annual US tax filings. If you didn't do that, it is considered cheating the IRS, and you might get asked where the money came from, and why you didn't report it before. You could still file FBARs for the previous years, and declare the reason as 'forgotten to file', that avoids you getting dinged with fees; but the question where the money came from is still open.
Generally, as a 'US person' your global income is taxable, so if you have money in a foreign country, it has to have come from somewhere, and that was probably a taxable event (or it was illegal).
I suppose it may also depend whether you are US or French tax resident...
I am UK tax resident but french national and I usually have to give a reason if I transfer money from UK to France.
I bet the french authorities have access to this and make use of it.