Can we derandomize subexponential algorithms given P=BPP?
Derandomization results transfer "upwards" using the technique of padding. Assume $\mathsf{P}=\mathsf{BPP}$. Suppose that some problem $A$ can be solved in randomized superpolynomial time $t(n)$ (which is time-constructible). Let $A'$ be the same problem, but with inputs padded to length $t(n)$. Then $A' \in \mathsf{BPP} = \mathsf{P}$. The following algorithm then solves $A$ in deterministic time $t(n)$: pad the input to length $t(n)$, then apply the polynomial time algorithm for $A'$.

Conversely, it is possible (given the current state of knowledge) that subexponential time algorithms can be randomized, but polynomial time algorithms can't (and it might be possible to construct a relativized world in which this happens unconditionally). That is, results don't transfer "downwards".