A traditional sieve gives a bound on the number of integers $n$ in an interval (say $I=[0,N]$) such that $$n\not\in S_p \mod p$$ for every prime $p$ in a set $\mathcal{P}$, where $S_p\subset \mathbb{Z}/p\mathbb{Z}$.
What happens if instead we are given a set $\mathcal{M}$ of composite moduli, and asked to give an upper bound on the number of integers $n$ in $I$ such that $$n\not\in S_m \mod m\;\;\; \forall m\in \mathcal{M},$$ where $S_m\subset \mathbb{Z}/m\mathbb{Z}$? Is there any literature on the matter? Note the elements of $\mathcal{M}$ need not be pairwise coprime.