This is perhaps more of a comment than an answer.

In various texts by Günter Harder he uses the terminology that *the Manin-Drinfeld principle holds* for the Shimura variety $X$, the coefficient system $\mathbb V$ and the cohomology theory $H(-)$ (usually $\ell$-adic cohomology or mixed Hodge theory) if there is a direct sum decomposition
$$ H(X,\mathbb V) = H_!(X,\mathbb V) \oplus H_{Eis}(X,\mathbb V)$$
into *inner* and *Eisenstein* cohomology. The former is the image of compactly supported cohomology in ordinary cohomology, the latter is the cokernel.

However I don't actually know any general result ensuring that the Manin-Drinfeld principle holds - but this may very well be due to my own ignorance. But you could start looking through his book (Eisensteinkohomologie und die Konstruktion gemischter Motive) as well as the various manuscripts on his webpage. You will at least find plenty of stuff of the form "if the Manin-Drinfeld principle holds, then we also know that..."