  We generalize the notion of asymptotic mapping class groups and allow them to
surject to the Higman--Thompson groups, answering a question of Aramayona and
Vlamis in the case of the Higman--Thompson groups. When the underlying surface
is a disk, these new asymptotic mapping class groups can be identified with the
ribbon and oriented ribbon Higman--Thompson groups. We use this model to prove
that the ribbon Higman--Thompson groups satisfy homological stability,
providing the first homological stability result for dense subgroups of big
mapping class groups. Our result can also be treated as an extension of
Szymik--Wahl's work on homological stability for the Higman--Thompson groups to
the surface setting.
