  We consider billiards with several possibly non-isometric and asymmetric
cusps at flat points; the case of a single symmetric cusp was studied
previously in Zhang (2017) and Jung & Zhang (2018). In particular, we show that
properly normalized Birkhoff sums of H\"older observables, with respect to the
billiard map, converge in Skorokhod's $M_1$-topology to an $\alpha$-stable
L\'evy motion, where $\alpha$ depends on the `curvature' of the flattest points
and the skewness parameter $\xi$ depends on the values of the observable at
those same points. Previously, Jung & Zhang (2018) proved convergence of the
one-point marginals to totally skewed $\alpha$-stable distributions for a
single symmetric cusp. The limits we prove here are stronger, since they are in
the functional sense, but also allow for more varied behaviour due to the
presence of multiple cusps. In particular, the general limits we obtain allow
for any skewness parameter, as opposed to just the totally skewed cases. We
also show that convergence in the stronger $J_1$-topology is not possible.
