  In this paper we study quotients of Lie algebroids and groupoids endowed with
compatible differential forms. We identify Lie theoretic conditions under which
such forms become basic and characterize the induced forms on the quotients. We
apply these results to describe generalized quotient and reduction processes
for (twisted) Poisson and Dirac structures, as well as to their integration by
(twisted, pre-)symplectic groupoids. In particular, we recover and generalize
several known results concerning Poisson reduction.
