  The "clustered chromatic number" of a class of graphs is the minimum integer
$k$ such that for some integer $c$ every graph in the class is $k$-colourable
with monochromatic components of size at most $c$. We prove that for every
graph $H$, the clustered chromatic number of the class of $H$-minor-free graphs
is tied to the tree-depth of $H$. In particular, if $H$ is connected with
tree-depth $t$ then every $H$-minor-free graph is $(2^{t+1}-4)$-colourable with
monochromatic components of size at most $c(H)$. This provides the first
evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about
defective colouring of $H$-minor-free graphs. If $t=3$ then we prove that 4
colours suffice, which is best possible. We also determine those minor-closed
graph classes with clustered chromatic number 2. Finally, we develop a
conjecture for the clustered chromatic number of an arbitrary minor-closed
class.
