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We discuss the low-energy limit of three-orbital Kondo-lattice and Hubbard models describing orbitals on a triangular lattice near half-filling. We analyze how very flat single-particle bands with nontrivial topological character, a Chern number , arise both in the limit of infinite on-site interactions as well as in more realistic regimes. Exact diagonalization is then used to investigate an effective one-orbital spinless-fermion model at fractional fillings including nearest-neighbor interaction ; it reveals signatures of fractional Chern insulator (FCI) states 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 FCI states are robust against disorder in the underlying magnetic texture that defines the topological character of the band. We also investigate competition between a FCI state and a charge …