I know how to prove that surjectivity is stable under base-change, i.e. that if $f:X\to S$ is a surjective morphism of schemes and we base-change via $g:S'\to S$ then $f_{S'}:X\times_SS'\to S'$ is surjective. The proofs I have seen amount to finding an element of $\text{Spec }k(s')\otimes_{k(s)}k(x)$ where $s'$ is some point whose fiber we want to show is non-empty and $f(x)=g(s')=s$.
This approach seems rather abstract to me and I wonder if $X$ and $S$ are affine if something more explicit may be said (or perhaps the above is explicit and I just don't understand what's really going on). Namely, I want to know if we have $\phi:A\to B$ and $\varphi:A\to C$, given some $\mathfrak{p}\in \text{Spec }C$ such that $\varphi^{-1}(\mathfrak{p})=\phi^{-1}(\mathfrak{q})$ for some $\mathfrak{q}\in\text{Spec }A$ can you describe as explicitly as possible the prime ideals of $B\otimes_A C$ lying over $\mathfrak{p}$ perhaps in terms of $\mathfrak{p}$ and $\mathfrak{q}$?