Graphs of bounded cliquewidth are polynomially $\chi$-bounded

We prove that if $\mathcal{C}$ is a hereditary class of graphs that is polynomially $\chi$-bounded, then the class of graphs that admit decompositions into pieces belonging to $\mathcal{C}$ along cuts of bounded rank is also polynomially $\chi$-bounded. In particular, this implies that for every positive integer $k$, the class of graphs of cliquewidth at most $k$ is polynomially $\chi$-bounded.