We study the stability of composite fermion fractional quantum Hall states in Harper-Hofstadter bands with Chern number |C|>1. From composite fermion theory, states are predicted to be found at filling factors ν=r/(kr|C|+1),r∈Z, with k=1 for bosons and k=2 for fermions. Here, we closely analyze these series in both cases, with contact interactions for bosons and nearest-neighbor interactions for (spinless) fermions. In particular, we analyze how the many-body gap scales as the bands are tuned to the effective continuum limit of Chern number |C| bands, realized near flux density n$$_ϕ$$=1/|C|. Near these points, the Hofstadter model requires 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 for these lattice geometries, we show that for both bosons and fermions, the vast majority of finite-size spectra yield the ground-state degeneracy predicted by composite fermion theory. For the chosen interactions, we confirm that states with filling factor ν=1/(k|C|+1) are the most robust and yield a clear gap in the thermodynamic limit. For bosons with contact interactions 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 stabilized for fermions with nearest-neighbor interactions in |C|>1 bands. For cases with a clear gap, we confirm that the thermodynamic limit commutes with the effective continuum limit within finite-size error bounds. We analyze the nature of the correlation functions for the Abelian composite fermion states and find that the correlation functions for |C|>1 states are smooth functions for positions separated by |C| sites along both axes, giving rise to |C|$$^2$$ sheets; some of which can be related by inversion symmetry. We also comment on two cases which are associated with a bosonic integer quantum Hall effect (BIQHE): For ν=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 ν=1 in |C|=2 bands, we present additional data confirming the existence of a BIQHE state.