IA living trust is a legal entity separate and apart from the individuals who fund the trust (the settlors). In order to transfer property into the US trust, the deed would need to be in the name of the trust, versus simply being in the individuals' names. For example, it would need to be titled as "John Doe, as Trustee of the Doe Family Trust". In most states, that can be as simple as executing a grant deed or a quit claim deed. Many people prefer to transfer say, Arizona real estate to their New York trust, in order to avoid ancillary probate in Arizona. Since this involves a foreign country, you would need to determine what their rules are regarding deeds - whether they will allow a trust to hold property; particularly a foreign trust (some countries have rules against foreign real property ownership).
Being a US attorney, I can only provide information regarding the American aspect of the question. For example, some countries have stamp duties (document tax) for transferring property to a trust, yet if one waits to bequeath it via a will, there may be no such stamp duties.
Furthermore, depending on citizenship, there are a myriad of tax issues. The US and France has a treaty - I will link you to it: http://ambafrance-us.org/IMG/pdf/french_us_estate_tax_treaty.pdf
which explains in detail the various situatiions as to which country is entitled to the estate tax (the purpose of the treaty is to avoid double taxation; the US has similar treaties with many countries).
Also, many countries have different estate taxes depending on whether the person who is inheriting the property is a blood relative or spouse, in addition to forced heirship laws .
Keep in mind, also, that the living trust is a product of US law, designed to help with US tax avoidance, and more importantly, to assist with avoiding the costs/fees/time of probate proceedings. Typically, US documents are not binding on other countries, particularly real property.
In situations such as this, it is advisable to consult with an international law firm, one that has attorneys licensed in both the United States and in France, so they can discuss in detail the pros/cons of each situation based on your particular circumstances.
You can make an inquiry into the American Bar Association to see if they have a referral program for cases such as this.http://apps.americanbar.org/legalservices/findlegalhelp/home.cfm
I hope this information helps explain the basics, and more importantly encourages you to seek individual counsel on this, in order to take advantage of the laws of both countries (especially since probate in France is notorious for taking much longer than a US probate).