Let $\mu=(\delta_A+\delta_B)/2$. Then the claim of Arnold - Krylov is the weak convergence of the convolutions $\mu^{*n}*\delta_x$ to the rotation invariant probability measure on the sphere (where $\mu^{*n}$ is the $n$-th convolution power of the probability measure $\mu$ on the group of rotations). A general answer to this question had been given by Stromberg Stromberg (1960) several years before Arnold - Krylov (they were not aware of this work) and largely goes back to Kawada - Ito (1940). According to Stromberg's Main Theorem, the sequence of convolution powers of a probability measure $\mu$ on a compact group $G$$K$ weakly converges to the Haar measure $m_K$ if and only if the support of $\mu$ is not contained in a coset of a proper closed normal subgroup.
EDIT The condition on the support of $\mu$ is obviously necessary as otherwise (if $\mu(gH)=1$ for a proper closed normal subgroup $H\subset K$) the image of $\mu$ under the quotient map $G\to G/H$ is concentrated on a single element, and therefore the image of $\mu^{*n}$ is the $n$-th power of this element. As for the original question, the point is that the group of isometries of a compact Riemannian manifold is also compact. There does exist "Haar measure" (or, rather, measures) on general compact manifolds - these are the unique invariant measures on the orbits of the group of isometries. Stromberg's theorem describes the limits of convolution powers on the group of isometries, and therefore, on its homogeneous spaces as well.
If you are interested in the convergence of the Cesaro averages rather than convolution powers themselves, condition $\mu(gH)<1$ is no longer necessary, and $(\mu +\dots+ \mu^{*n})/n$ converges to the Haar measure on the subgroup generated by the support for any measure $\mu$ on a compact group $K$. All this essentially goes back to Ito -- Kawada, and a good account can be found in old Grenander's book.