On the base size and minimal degree of transitive groups

Let $G$ be a permutation group, and denote with $\mu (G)$ and $b(G)$ its minimal degree and base size respectively. We show that for every $\epsilon >0$, there exists a transitive permutation group $G$ of degree $n$ with $$\mu (G)b(G)\ge n^{2-\epsilon }\mathrm{.}$$ We also identify some classes of transitive and intransitive groups whose base size and minimal degree have a smaller upper bound, shared with primitive groups.