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 $\varepsilon>0$, there exists a transitive permutation group $G$ of degree $n$ with $$\mu(G)b(G) \geq n^{2-\varepsilon}.$$ 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.