A family ${\cal F}$ of graphs is asymptotically $χ$-bounded with bounding function $f$ if almost every graph $G$ in the family satisfies $χ(G) \le f(ω(G))$. A graph is $H$-free if it does not contain $H$ as an induced subgraph. We ask which hereditary families are asymptotically $χ$-bounded, and discuss some related questions. We show that for every tree $T$, almost all $T$-free graphs $G$ satisfy $χ(G)=ω(G)$. We show that for every cycle $C_k$ except $C_6$, almost every $C_k$-free graph $G$ satisfies $χ(G) = ω(G)$. We show that the $C_6$-free graphs are asymptotically $χ$-bounded with bounding function $f(w)=(1+o(1))\frac{w^2}{\log w}$.