The Collins-Soper (CS) kernel is a nonperturbative function that characterizes the rapidity evolution of transverse-momentum-dependent parton distribution functions (TMDPDFs) and wave functions. In this paper, we calculate the CS kernel for pion and proton targets and for quasi-TMDPDFs of leading and next-to-leading power. The calculations are carried out on the CLS ensemble H101 with dynamical Nf=2+1 clover-improved Wilson fermions. Our analyses demonstrate the consistency of different lattice extractions of the CS kernel for mesons and baryons, as well as for twist-two and twist-three operators, even though lattice artifacts could be significant. This consistency corroborates the universality of the lattice-determined CS kernel and suggests that a high-precision determination of it is in reach.