Let $\mathsf{k}$ be a field, with $A$ some $\mathsf{k}$-algebra, and $M$ some finite dimensional $A$-module. Then $M$ admits a composition series
$$ 0 = M_0 \subset M_1 \subset \dots \subset M_\ell = M, $$
where the composition factors $M_{i+1} / M_i$ are simple modules. Suppose we pick some index $0 \leq k \leq \ell$. Then $\overline{M}_k = M / M_k$ is once again a finite dimensional $A$-module, and so has some composition factors.
Is is true that the isomorphism classes of these composition factors of $\overline{M}_k$ are given by
$$ M_{k+1} / M_k,\quad M_{k+2} / M_{k+1}, \quad \dots \quad , M_{\ell} / M_{\ell-1} $$
Going to answer this question myself since I think I have solved it.
Notice that
$$ 0 = M_k / M_k \subset M_{k+1} / M_k \subset \dots \subset M / M_k, $$
is a composition series for $M / M_k$, and so the composition factors of $M / M_k$ are $$ \left( M_{i+1} / M_k \right) / \left( M_{i} / M_k \right) \cong M_{i+1} / M_i,$$ for $i \geq k$, and so we're done.