  In this paper we study two classes of meromorphic functions previously
studied by Mayer, Kotus, and Urba\'nski. In particular we estimate a lower
bound for the Julia set and the set of escaping points for non-autonomous
additive and affine perturbations of functions from these classes. For
particular classes we are able to calculate these dimensions exactly. In these
cases, we are able to reinterpret our results to show that the Hausdorff
dimension of the set of points in the Julia set which escape to infinity is
stable under sufficiently small additive perturbations. We accomplish this by
constructing non-autonomous iterated function systems, whose limit sets sit
inside of the aforementioned non-autonomous Julia sets. We also give estimates
for the eventual and eventual hyperbolic dimensions of these non-autonomous
perturbations.
