  In this paper, we study pushforwards of log pluricanonical bundles on
projective log canonical pairs $(Y,\Delta)$ over the complex numbers, partially
answering a Fujita-type conjecture due to Popa and Schnell in the log canonical
setting. We show two effective global generation results. First, when $Y$
surjects onto a projective variety, we show a quadratic bound for generic
generation for twists by big and nef line bundles. Second, when $Y$ is fibered
over a smooth projective variety, we show a linear bound for twists by ample
line bundles. These results additionally give effective non-vanishing
statements. We also prove an effective weak positivity statement for log
pluricanonical bundles in this setting, which may be of independent interest.
In each context we indicate over which loci positivity holds. Finally, using
the description of such loci, we show an effective vanishing theorem for
pushforwards of certain log-sheaves under smooth morphisms.
