The possibility of realizing lattice analogs of fractional quantum Hall (FQH) states, so-called fractional Chern insulators (FCIs), in nearly flat topological (Chern) bands has attracted a lot of recent interest. Here, we make the connection between Abelian as well as non-Abelian FQH states and FCIs more precise. Using a gauge-fixed version of Qi's Wannier basis representation of a Chern band, we demonstrate that the interpolation between several FCI states, obtained by short-range lattice interactions in a spin-orbit-coupled kagome lattice model, and the corresponding continuum FQH states is smooth: the gap remains approximately constant and extrapolates to a finite value in the thermodynamic limit, while the low-lying part of the orbital entanglement spectrum remains qualitatively unaltered. The orbital entanglement spectra also provide a first glimpse of the edge physics of FCIs via the bulk-boundary correspondence. Corroborating these results, we find that the squared overlaps between the FCI and FQH ground states are as large as 98.7% for the 8-electron Laughlin state at ν=13 (consistent with an earlier study) and 97.8% for the 10-electron Moore-Read state at ν=12. For the bosonic analogs of these states, the adiabatic continuity is also shown to hold, albeit with somewhat smaller associated overlaps, etc. Although going between the Chern bands to the Landau-level problem is often smooth, we show that this is not always the case by considering fermions at filling fraction ν=45, where the interpolation between Hamiltonians describing the two systems results in a phase transition.