Question: For standard BBN calculations, we use the baryon to photon ratio
$$
\eta=\frac{n\_b}{n\_\gamma}
$$
that we get from CMB. Now, this clearly assumes that
$$
\eta\_{BBN}=\eta\_{CMB}.
$$
There are around 400000 years in between. Why can we assume that $\eta$ stays the same until then? I thought $\eta$ goes inversely with the scale factor $a$.

Answer: 
The baryon to photon ratio is essentially fixed by the annihilation of particles and anti-particles in the early universe. If there had been absolute symmetry between the numbers of particles and anti-particles, then the ratio would have been zero! Once all the particle/anti-particle annihilation processes have concluded (once photon energies fell below the rest mass of an election-positron pair), then photons outnumbered baryons and electrons by a factor of $\sim 10^9$.

The (ionised) matter and radiation at that point are in equilibrium. Thus for every process (e.g. thermal bremsstrahlung) producing a photon, there is a similar rate of the inverse process occurring that removes a photon. This effectively conserves the photon number in a comoving volume. Obviously as expansion takes place, the *density* of photons decreases, but the baryon density also decreases by exactly the same factor and thus their ratio, $\eta$, is constant.

Then, even when the radiation-matter equilibrium is lost, and the electrons combine with protons and alpha particles to form hydrogen and helium atoms; the photon number is still approximately conserved because the combination processes can only increase the photon numbers by about 1 part in a billion.
I have no clear idea why you ask this question, and what information you are looking for. By baryons I am guessing you mean protons and neutrons. I am also assuming you are ignoring dark matter. Protons and neutrons are frequently combining with electrons to form atoms. In most of the matter in the universe (ignoring dark matter) the vast bulk of it is in galaxies. The gravity effects within galaxies create interactions in with protons and neutrons in the form of neuclei interact with electrons, and a lot of photons are created. So it is NOT assumed that the ratio between baryons and photons are constant. The number of baryons are almost entirely unchanged, but photons are created frequently which do not make further interactions, not at least in any long period of time. It is conceivable that black holes (Hawking Radiation) may have some role to play, which at some very extreme future time, all of the baryons and photons may become so far apart that they will no longer interact, so then the ratio would be constant.