  Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially
accessible domain (aka uniform domain), i.e., a set which satisfies the
interior Corkscrew and Harnack chain conditions, respectively
scale-invariant/quantitative versions of openness and path-connectedness.
Assume that $\Omega$ satisfies the so-called capacity density condition. Let
$L_0u=-\mathrm{div}(A_0\nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$ be two real
(non-necessarily symmetric) uniformly elliptic operators, and write
$\omega_{L_0}$, $\omega_L$ for the associated elliptic measures. The goal of
this program is to find sufficient conditions guaranteeing that $\omega_L$
satisfies an $A_\infty$-condition or a $RH_q$-condition with respect to
$\omega_{L_0}$. We show that if the discrepancy of the two matrices satisfies a
natural Carleson measure condition with respect to $\omega_{L_0}$, then
$\omega_L\in A_\infty(\omega_{L_0})$. Moreover, $\omega_L\in
RH_q(\omega_{L_0})$ for any given $1<q<\infty$ if the Carleson measure
condition is assumed to hold with a sufficiently small constant. This extends
previous work of Fefferman-Kenig-Pipher and Milakis-Pipher-Toro who considered
Lipschitz and chord-arc domains. Here we go beyond as the capacity density
condition is much weaker than the existence of exterior Corkscrew balls. The
"large constant" case, where the discrepancy satisfies a Carleson measure
condition, is new even for nice domains such as the unit ball, the upper
half-space, or Lipschitz domains, and is obtained using the method of
extrapolation of Carleson measure. Our domains do not have a nice surface
measure: all the analysis is done with the underlying measure $\omega_{L_0}$.
When particularized to Lipschitz, chord-arc, or 1-sided chord-arc domains, we
recover previous results and extend some of them. Our arguments rely on the
square function and non-tangential estimates proved in arXiv:2103.10046.
