  We study the stability of composite fermion fractional quantum Hall states in
Harper-Hofstadter bands with Chern number $|C|>1$. We analyze the states of the
composite fermion series for bosons with contact interactions and (spinless)
fermions with nearest-neighbor interactions. We examine the scaling of the
many-body gap as the bands are tuned to the effective continuum limit
$n_\phi\to 1/|C|$. Near these points, the Hofstadter model realises large
magnetic unit cells that yield bands with perfectly flat dispersion and Berry
curvature. We exploit the known scaling of energies in the effective continuum
limit in order to maintain a fixed square aspect ratio in finite-size
calculations. Based on exact diagonalization calculations of the band-projected
Hamiltonian, we show that almost all finite-size spectra yield the ground-state
degeneracy predicted by composite fermion theory. We confirm that states at low
ranks in the composite fermion hierarchy are the most robust and yield a clear
gap in the thermodynamic limit. For bosons in $|C|=2$ and $|C|=3$ bands, our
data for the composite fermion states are compatible with a finite gap in the
thermodynamic limit. We also report new evidence for gapped incompressible
states of fermions in $|C|>1$ bands, which have large entanglement gaps. For
cases with a clear spectral gap, we confirm that the thermodynamic limit
commutes with the effective continuum limit. We analyze the nature of the
correlation functions for the Abelian composite fermion states and find that
they feature $|C|^2$ smooth sheets. We examine two cases associated with a
bosonic integer quantum Hall effect (BIQHE): For $\nu=2$ in $|C|=1$ bands, we
find a strong competing state with a higher ground-state degeneracy, so no
clear BIQHE is found in the band-projected Hofstadter model; for $\nu=1$ in
$|C|=2$ bands, we present additional data confirming the existence of a BIQHE
state.
