In short: There is absolutely no such relations, and in general essentially no property of arrows of $C$ (except being a split epi/split mono or an iso) have any effect on the base change functor.

Indeed, because of Grothendieck's construction, absolutely any (pseudo)-functor from $X^{op}$ to $Cat$ is obtained as the base change functor corresponding to a Fibration of codomain $X$.

So you can send you epis and mono on whatever you like, you are asking if a completely general functor send epi or mono to some specific classes of maps. The only thing that are preserved are those that are expressed using justs "equations", so: "isomorphisms", "split epis" and "split monos" and that is essentially it.

In short: There is absolutely no such relations, and in general essentially no properties of arrows of the codomain (except being a split epi/split mono or an iso) have any effect on the base change functor.

Indeed, because of Grothendieck's construction, absolutely any (pseudo)-functor from $X^{op}$ to $Cat$ is obtained as the base change functor corresponding to a Fibration of codomain $X$.

So you can send you epis and mono on whatever you like, you are asking if a completely general functor send epi or mono to some specific classes of maps. The only thing that are preserved are those that are expressed using justs "equations", so: "isomorphisms", "split epis" and "split monos" and that is essentially it.