From the BO approximation, we have the product of the electronic \varphi_i(r,R) and the nuclear \eta_i(R) wavefunction. For the transition, we use Fermi's Golden rule, where the Dipole-Operator \mu "initiates" the transition. So we end up in
r_{i\rightarrow j}=\left\langle \eta_i(R) \varphi_i(r,R) |\mu| \varphi_j(r,R) \eta_j(R) \right\rangle.
Here we have an inner integral over the electron coordinates r and an outer integral over the nuclei coordinates R. It is important to note here that the inner integral \left\langle \varphi_i(r,R) |\mu| \varphi_j(r,R) \right\rangle is a function of R. The approximation is now that this inner integral is taken out of the outer interal, even though the former one is dependent of R - which is the integration variable of the outer integral. Now the above equation looks like this:
r_{i\rightarrow j}= \left\langle\varphi_i(r,R) |\mu| \varphi_j(r,R) \right\rangle \cdot\left\langle \eta_i(R)|\eta_j(R) \right\rangle
So actually the electronic integral is handled independently of the nuclei integral. The former one is a usual transition (with an operator for the transition according to Fermi's Golden rule), while the latter one is only an overlap of wavefunctions any more!
