  In this manuscript, we elaborate on a procedure to derive
$\epsilon$-factorised differential equations for multi-scale, multi-loop
classes of Feynman integrals that evaluate to special functions beyond multiple
polylogarithms. We demonstrate the applicability of our approach to diverse
classes of problems, by working out $\epsilon$-factorised differential
equations for single- and multi-scale problems of increasing complexity. To
start we are reconsidering the well-studied equal-mass two-loop sunrise case,
and move then to study other elliptic two-, three- and four-point problems
depending on multiple different scales. Finally, we showcase how the same
approach allows us to obtain $\epsilon$-factorised differential equations also
for Feynman integrals that involve geometries beyond a single elliptic curve.
