# Is combinatory strong reduction equivalent to lambda beta-eta reduction?

I'm familiar with the correspondence between combinatory logic and $\lambda$-calculus at equality level, ie $=_{\beta\eta}$ corresponds to $=_s$ (the equivalence relation generated by strong reduction) and that there is also a combinatory equality $=_{C\beta}$ corresponding to $=_\beta$ in $\lambda$.

However, I'm having trouble understanding what happens at reduction level. In particular, is it true that Curry's strong reduction corresponds to $\rightarrow_{\beta\eta}$? For instance, could we use strong reduction to "evaluate" (up to $\beta\eta$) $\lambda$-terms? Also, if I understand correctly, is it even harder to come up with a combinatory reduction equivalent to $\rightarrow_\beta$?