  We investigate how the dimensionless ratio given by the Chern-Simons
diffusion rate $\Gamma_{\textrm{CS}}$ divided by the product of the entropy
density $s$ and temperature $T$ behaves across different kinds of phase
transitions in the class of bottom-up non-conformal Einstein-dilaton
holographic models originally proposed by Gubser and Nellore. By tuning the
dilaton potential, one is able to holographically mimic a first order, a second
order, or a crossover transition. In a first order phase transition,
$\Gamma_{\textrm{CS}}/sT$ jumps at the critical temperature (as previously
found in the holographic literature), while in a second order phase transition
it develops an infinite slope. On the other hand, in a crossover,
$\Gamma_{\textrm{CS}}/sT$ behaves smoothly, although displaying a fast
variation around the pseudo-critical temperature. In all the cases,
$\Gamma_{\textrm{CS}}/sT$ increases with decreasing $T$. The behavior of the
Chern-Simons diffusion rate across different phase transitions is expected to
play a relevant role for the chiral magnetic effect around the QCD critical end
point, which is a second order phase transition point connecting a crossover
band to a line of first order phase transition. Our findings in the present
work add to the literature the first predictions for the Chern-Simons diffusion
rate across second order and crossover transitions in strongly coupled
non-conformal, non-Abelian gauge theories.
