  We study the forces and optical pumping within grating magneto-optical traps
(MOTs) operating on transitions with non-trivial level structure. In contrast
to the standard six-beam MOT configuration, rate equation modelling predicts
that the asymmetric laser geometry of a grating MOT will produce spin-polarized
atomic samples. Furthermore, the Land\'e $g$-factors and total angular momenta
of the trapping transition strongly influence both the confinement and
equilibrium position of the trap. Using the intuition gained from the rate
equation model, we realize a grating MOT of fermionic $^{87}$Sr and observe
that it forms closer to the center of the trap's quadrupole magnetic field than
its bosonic counterpart. We also explore the application of grating MOTs to
molecule laser cooling, where the rate equations suggest that dual-frequency
operation is necessary, but not sufficient, for stable confinement for type-II
level structures. To test our molecule laser cooling models, we create grating
MOTs using the $D_1$ line of $^7$Li and see that only two of the four possible
six-beam polarization configurations operate in the grating geometry. Our
results will aid the development of portable atom and molecule traps for time
keeping, inertial navigation, and precision measurement.
