Yes, they're thermalized and captured. Remember that the free neutron lifetime is roughly fifteen minutes, and that thermal neutrons are moving a couple kilometers per second. The chance of going fifteen minutes without a capture interaction is tiny. (Even in so-called "neutron bottle traps," it's hard to get the mean time before neutron disappearance to approach the beta-decay lifetime.) There'll be some free decay, but not much.
The atmosphere has lots of water vapor, and the ground contains lots of water, for capture on hydrogen; the cross sections for capture on carbon and oxygen are pretty low (they're alpha-particle nuclei) but the cross sections on nitrogen (1.9 barn, for thermal neutrons) and silicon (0.17 barn) are comparable to hydrogen (0.33 barn). Iron's in the same territory. You can browse the Neutron Data Booklet as well as I can. Odd-proton nuclei, like copper, are typically unstable after neutron capture and will beta-decay after some time.
Actually what I remember reading is that more cosmic ray neutrons are produced in the ground, by fast muon spallation, than in processes like $\mathrm{^{14}N(p,n)^{14}C}$ in the atmosphere. The first data appeared in JETP, the Soviet journal, and mentioned that the thermal flux at ground level was markedly increased by the presence of a half-meter of snow to act as a moderator.
