Is there a corollary of Hurwitz theorem in complex analysis when the sequence of holomorphic functions depends also of a real parameter?

Question

Let us consider a sequence of holomorphic functions $f_{\varepsilon}(z,a)$ where $\varepsilon>0$ and $a$ a real parameter. We can find explicitly the roots in the limiting case $f_0(z,a)=0$, as functions of the parameter $a$. Now, we are interested in a ''perturbation'' of the limiting case when $\varepsilon>0$. More generally, is there a corollary or variant of Hurwitz theorem in the case of a parametrized sequence of holomorphic functions?