  Let $G$ be a group and let $S=\bigoplus_{g \in G} S_g$ be a $G$-graded ring.
Given a normal subgroup $N$ of $G$, there is a naturally induced $G/N$-grading
of $S$. It is well-known that if $S$ is strongly $G$-graded, then the induced
$G/N$-grading is strong for any $N$. The class of epsilon-strongly graded rings
was recently introduced by Nystedt, \"Oinert and Pinedo as a generalization of
unital strongly graded rings. We give an example of an epsilon-strongly graded
partial skew group ring such that the induced quotient group grading is not
epsilon-strong. Moreover, we give necessary and sufficient conditions for the
induced $G/N$-grading of an epsilon-strongly $G$-graded ring to be
epsilon-strong. Our method involves relating different types of rings equipped
with local units (s-unital rings, rings with sets of local units, rings with
enough idempotents) with generalized epsilon-strongly graded rings.
