We examine sparse graph automorphism groups from the perspective of the Kechris-Pestov-Todorčević (KPT) correspondence. The sparse graphs that we discuss are Hrushovski constructions: we consider the 'ab initio’ Hrushovski construction M_0, the Fraïssé limit of the class of 2-sparse graphs with self-sufficient closure; M_1, a simplified version of M_0; and the ω-categorical Hrushovski construction M_F. We prove a series of results that show that the automorphism groups of these Hrushovski constructions demonstrate very different behaviour to previous classes studied in the KPT context. Extending results of Evans, Hubička and Nešetřil, we show that Aut(M_0) has no coprecompact amenable subgroup. We investigate the fixed points on type spaces property, a weakening of extreme amenability, and show that for a particular choice of control function F, Aut(M_F) does not have any closed oligomorphic subgroup with this property. Next we consider the Aut(M_1)-flow of linear orders on M_1, and show that minimal subflows of this have all Aut(M_1)-orbits meagre. We give partial analogous results for the Aut(M_0)-flow of linear orders on M_0, and find the universal minimal flow of the automorphism group of the “dimension 0” part of M_0.