  We consider the Cauchy problem for homogeneous linear
$q$-difference-differential equations with constant coefficients. We
characterise convergent, $k$-summable and multisummable formal power series
solutions in terms of analytic continuation properties and growth estimates of
the Cauchy data. We also introduce and characterise sequences preserving
summability, which make a very useful tool, especially in the context of moment
differential equations.
