What is the meaning of a number changing process?

On my lecturer's notes on Dark Matter I am told:

"It is customary to define densities normalised by the time dependent volume $$V (t) = a(t)^3$$. The reason for this is that, in the absence of number changing processes, the comoving number density remains constant with time evolution."

I don't understand what is meant by a number changing process.

The image is given as an example of the preservation of the comoving number density.

Initially I thought a number changing process was one in which the number of the particles changed as the process evolved but I believe the image is trying to emphasise that the number of particles is kept constant.

You are right. The number of particles does not change, but the volume increases as $$V(t) = V_0 a^3(t)$$, because of expansion. Naively one can compute the number density as $$n(t) = \frac{N(t)}{V(t)}=\frac{N_0}{V_0 a^3(t)},$$ where I used the fact that $$N(t)=N_0$$ in the absence of process that change the number of particles.
However, if we spice up things by normalizing the volume by the expansion factor $$a^3(t)$$ in the previous expression, we have $$\hat{n}(t) = \frac{N_0}{V_0 a^3(t)/a^3(t)}= \frac{N_0}{V_0},$$ which is indeed the definition of comoving number density and it is manifestly time-independent.