Being locally free is a property of quasi-coherent modules which does not descend in the fpqc topology (see Remark Tag 05VF). But what happens for fppf coverings? More precisely we ask:
Suppose $A \to B$ is a faithfully flat ring map of finite presentation. Let $M$ be an $A$-module such that $M \otimes_A B$ is free. Is $M$ a locally free $A$-module?
Note that the answer is yes for finite modules. The problem is with "large" modules. Also, if $A$ is Noetherian, then the answer is yes. This follows from results of Bass. But in general, question_bot doesn't know the answer. Do you?