This is probably a difficult question. I would like to understand some particular cases and get some references. The rough question is the following:

Let $X$ be a PEL Shimura variety and $\pi: \mathcal{A} \to X$ the universal abelian scheme over $X$.

What can be said about the monodromy of the local system $R^1\pi_\ast \mathbb{Q}$?

For instance, if $Y(N)$ is the modular curve of level $N$ and $\mathcal{E}$ the corresponding universal elliptic curve and we consider the compactification $Y(N) \hookrightarrow X(N)$, what is the local monodromy around a cusp?

Has this computation been done for Hilbert modular varieties?