  The goal of this paper is twofold. On the one hand, we introduce a
quasi-homogeneous version of the classical ideal MHD system and study its
well-posedness in critical Besov spaces $B^s_{p,r}(\mathbb{R}^d)$, $d\geq2$,
with $1<p<+\infty$ and under the Lipschitz condition $s>1+d/p$ and
$r\in[1,+\infty]$, or $s=1+d/p$ and $r=1$. A key ingredient is the
reformulation of the system \textsl{via} the so-called Els\"asser variables. On
the other hand, we give a rigorous justification of quasi-homogeneous MHD
models, both in the ideal and in the dissipative cases: when $d=2$, we will
derive them from a non-homogeneous incompressible MHD system with Coriolis
force, in the regime of low Rossby number and for small density variations
around a constant state. Our method of proof relies on a relative entropy
inequality for the primitive system, and yields precise rates of convergence,
depending on the size of the initial data, on the order of the Rossby number
and on the regularity of the viscosity and resistivity coefficients.
