  Axionic holographic RG flow solutions are studied in the context of general
Einstein-Axion-Dilaton theories. A non-trivial axion profile is dual to the
(non-perturbative) running of the $\theta$-term for the corresponding instanton
density operator. It is shown that a non-trivial axion solution is incompatible
with a non-trivial (holographic) IR conformal fixed point. Imposing a suitable
axion regularity condition allows to select the IR geometry in a unique way.
The solutions are found analytically in the asymptotic UV and IR regimes, and
it is shown that in those regimes the axion backreaction is always negligible.
The axion backreaction may become important in the intermediate region of the
bulk. To make contact with the axion probe limit solutions, a systematic
expansion of the solution is developed. Several concrete examples are worked
out numerically. It is shown that the regularity condition always implies a
finite allowed range for the axion source parameter in the UV. This translates
into the existence of a finite (but large) number of saddle-points in the large
$N_c$ limit. This ties in well with axion-swampland conjectures.
