  In this paper we propose a general coorbit space theory suitable to define
coorbits of quasi-Banach spaces using an abstract continuous frame, indexed by
a locally compact Hausdorff space, and an associated generalized voice
transform. The proposed theory realizes a further step in the development of a
universal abstract theory towards various function spaces and their atomic
decompositions which has been initiated by Feichtinger and Gr{\"o}chenig in the
late 1980ies. We combine the recent approaches in Rauhut, Ullrich and Rauhut to
identify, in particular, various inhomogeneous (quasi-Banach) spaces of
Besov-Lizorkin-Triebel type. To prove the potential of our new theory we apply
it to spaces with variable smoothness and integrability which have attracted
significant interest in the last 10 years. From the abstract discretization
machinery we obtain atomic decompositions as well as wavelet frame isomorphisms
for these spaces.
