Locally free modules do not satisfy descent for fpqc coverings. I have an example involving a countable “product” of affine curves, which I will upload to the stacks project soon.

But what about fppf descent? Suppose A —> B is a faithfully flat ring map of finite presentation. Let M be an A-module such that M ⊗_A B is free. Is M a locally free A-module? (By this I mean locally free on the spectrum of A.) It turns out that if A is Noetherian, then the answer is yes. This follows from the results of Bass in his paper on “big” projective modules. But in general I don’t know the answer. If you do know the answer, or have a reference, please email me.

To show fppf descent, it would be enough to show descent along finite fppf morphisms and étale surjective morphisms. Whether one has descent in either of these two cases is not clear to me. One potential problem is the following:

If M is a locally free A-module of finite rank, then any finite subset Z of Spec(A) is contained in an open subset over which M is free (this follows from the fact that M is free over the semi-local ring given by localizing in Z). What about modules of infinite rank?

Asking for descent of local freeness along finite flat morphisms is essentially equivalent to the above question.