Let $X$ and $Y$ be smooth projective varieties over $\mathbb C$. Suppose I have given a line bundle $L$ on $X\times Y$ with a connection relative to $Y$, i.e.
$\nabla: L \rightarrow \Omega^1_{X\times Y/Y} \otimes L$.
One can push it down via the first projection $p$ and so gets a connection on $p_{*}L$ on $X$.
Furthermore let $V$ be any quasicoherent sheaf on $X \times Y$; then there is no canonical way to associate a connection to $L\otimes V$ relative to $Y$ from the given data, simply because $V$ doesn't have a connection. But please correct me if I am wrong here.
Question: But is there nevertheless a natural way to get a connection on $p_{*}(L\otimes V)$ from the data?