As a non-EU citizen, can work in all Schengen countries with a French work permit?

I am from a non-EU country. I have studied my Master's Degree in France and been living here for about 3 years. I got an offer (CDI) from an American company. In fact, it's a Graduate Programme for two years and I'll be working in 4-5 different countries during this period. My first assignment is in France, the rest to be determined but will probably be some other Schengen countries such as Germany, Belgium, Switzerland, etc. I'm soon gonna apply for the work permit here in France. The company will of course help me with the papers and so on but what I would like to know is whether a the French work permit I'll get will be valid to work let's say in Germany or in Belgium. The duration of the assignments are also to be determined, but I guess each one will be for about 3-5 months. I don't know whether I should apply for a simple work permit or something more comprehensive -if there is such a thing. I'll be glad if you share your opinions with me. Thanks in advance!