- He did not appreciate the men taking charge of the project. vs.
- He did not appreciate the men's taking charge of the project. If sentence 2 is technically correct it seems awfully silly and affected.
Though I would likely select example one, it dependes on your emphasis. This is often called topicalization. The second example using the possive shifts some of the emphasis of the noun phrase onto the "men (who) were taking charge" whereas the first example focuses more on the action of "men taking change". The reason for this shift is that a possessive word creates a more complex underlying syntactic structure which "lifts" the possessive into a controlling postion over the remainder of the NP. I think it's the lack of complexity that makes the first eample easier to read. But the topicalization is a legitimate tool in your writing arsenal if you need it.
You can explore the effects of the possive by doing substitution testing where you change the progressive tense phrasal verb "taking charge" and changing it to another verb. In some cases the exchange may be straight forward, but with some verbs you discover that the verb will force a change in structure. These strucural changes are often what creates the emphasis change.