  We discuss the low-energy limit of three-orbital Kondo-lattice and Hubbard
models describing $t_{2g}$ orbitals on a triangular lattice near half-filling.
We analyze how very flat bands with non-trivial topological character, a Chern
number C=1, arise both in the limit of infinite on-site interactions as well as
in more realistic regimes. Exact diagonalization is then used to investigate
fractional filling of an effective one-band spinless-fermion model including
nearest-neighbor interaction $V$; it reveals signatures of fractional Chern
insulators (FCIs) for several filling fractions. In addition to indications
based on energies, e.g. flux insertion and fractional statistics of quasiholes,
Chern numbers are obtained. It is shown that FCIs are robust against disorder
in the underlying magnetic texture that defines the topological character of
the band. We also investigate competition between FCI states and a charge
density wave (CDW) and discuss particle-hole asymmetry as well as Fermi-surface
nesting. FCI states turn out to be rather robust and do not require very flat
bands, but can also arise when filling or an absence of Fermi-surface nesting
disfavor the competing CDW. Nevertheless, very flat bands allow FCI states to
be induced by weaker interactions than those needed for more dispersive bands.
