  Meta-theorems for polynomial (linear) kernels have been the subject of
intensive research in parameterized complexity. Heretofore, meta-theorems for
linear kernels exist on graphs of bounded genus, $H$-minor-free graphs, and
$H$-topological-minor-free graphs. To the best of our knowledge, no
meta-theorems for polynomial kernels are known for any larger sparse graph
classes; e.g., for classes of bounded expansion or for nowhere dense ones. In
this paper we prove such meta-theorems for the two latter cases. More
specifically, we show that graph problems that have finite integer index (FII)
have linear kernels on graphs of bounded expansion when parameterized by the
size of a modulator to constant-treedepth graphs. For nowhere dense graph
classes, our result yields almost-linear kernels. While our parameter may seem
rather strong, we argue that a linear kernelization result on graphs of bounded
expansion with a weaker parameter (than treedepth modulator) would fail to
include some of the problems covered by our framework. Moreover, we only
require the problems to have FII on graphs of constant treedepth. This allows
us to prove linear kernels for problems such as Longest Path/Cycle, Exact
$s,t$-Path, Treewidth, and Pathwidth, which do not have FII on general graphs
(and the first two not even on bounded treewidth graphs).
