Speaking of royalty and museum collections; whatever reservations or thoughts royal courts of Europe and or their nobility had about events in revolutionary France, it didn't stop them from snapping up furniture and other possessions of the Bourbons.
An auction took place between 25 August 1793 and 11 August 1794. The furnishings and art of Versailles Palace, including the furniture, mirrors, baths and kitchen equipment, were sold in seventeen thousand lots by revolutionary government.
So keen was the revolutionary government to disperse and be rid of any memory of the Bourbons/autocratic government that they preferred to sell the furniture and furnishings of Versailles to foreigners. To sweeten the deal it was advertised in brochures or whatever that such buyers could take their purchases out of France tax free; they came in droves from all over the world.
Much of Buckingham palace contains furniture from Versailles (then Prince of Wales working via agents took away things by several ship loads).
A period poster advertising the sale of the furniture and effects of the person referred to as the "So-Called Queen" (Marie Antoinette) took place on Sunday, August 25, 1793 - seven months after the execution of her husband and two months before her own death.
Besides European royalty and aristocracy the Americans were there in force as well. In years since many families acquired more pieces via private sales or at auctions.
France has put in place laws against export of cultural or historic items such as art, furniture, etc.., but if someone doesn't know what they've got or looks at things carefully it can slip through.
Royal residences were inventoried in what is called the Journal du Garde Meuble for the years 1666-1792, and pieces from Versailles and elsewhere do have identifying markings. So in theory it should be easy to spot such things, but again not everyone examines things carefully. Others (such as antique dealers, auction houses, private sellers) fear if brought to attention of French government they will claim the thing as part of France's cultural heritage .
France has spent vast sums, and continues to spend; French government recently received a legacy of 20 million USD from an heiress with stipulation it only be spent on acquiring and returning former furniture of Versailles to the palace.
Making that job slightly easier is that fussy period dark antique furniture is falling out of fashion with younger generations. Museums in USA often already have storerooms or collections sufficient and don't want any more. IIRC the MET has put out the word "no more French furniture...".