I was wondering since I was reading about the Franco-Thai war, and don't really understand if the Thai were fighting the Nazi sympathizer French or the "good" French.
France in WW2 is very confusing.
In the case of Syria and Lebanon, Vichy France was in control. There was no need to "takeover." After the surrender of France, the Armee du Levant and the administration were pro-Vichy, and the Vichy government was able to further appoint its own administrator General Henri Dentz.
It's my understanding that the invasion of Syria/Lebanon has been woefully understudied in part because of the exceptionally high casualty rates—the Vichy French fought the "good" French and British invaders tooth and nail—in a way that was somewhat embarrassing for everyone.
As a result I don't know any particularly good sources on the question, but it's discussed in passing in, for example, D.K. Fieldhouse's Western Imperialism in the Middle East 1914-1958.