  Given a fibration $f$ between two projective manifolds $X$ and $Y$, we
discuss the effective generation of the higher direct images
$R^{i}f_{\ast}(K^{m}_{X})$, where $K^{m}_{X}$ is the $m$-th tensor power of the
canonical bundle of $X$. In particular, we answer two questions posed by
Popa--Schnell in [PS14].
