  We study orbits for rational equivalence of zero-cycles on very general
abelian varieties by adapting a method of Voisin to powers of abelian
varieties. We deduce that, for $k$ at least $3$, a very general abelian variety
of dimension at least $2k-2$ has covering gonality greater than $k$. This
settles a conjecture of Voisin. We also discuss how upper bounds for the
dimension of orbits for rational equivalence can be used to provide new lower
bounds on other measures of irrationality. In particular, we obtain a
strengthening of the Alzati-Pirola bound on the degree of irrationality of
abelian varieties.
