We analyze the kinematics of cosmological spacetimes with nonzero torsion, in the framework of the classical Einstein-Cartan gravity. After a brief introduction to the basic features of spaces with nonvanishing torsion, we consider a family of observers moving along timelike worldlines and focus on their kinematic behavior. In so doing, we isolate the irreducible variables monitoring the observers' motion and derive their evolution formulas and associated constraint equations. Our aim is to identify the effects of spacetime torsion, and the changes they introduce into the kinematics of the standard, torsion-free, cosmological models. We employ a fully geometrical approach, imposing no restrictions on the material content, or any a priori couplings between torsion and spin. Also, we do not apply the familiar splitting of the equations, into a purely Riemannian component plus a torsion/spin part, at the start of our study, but only introduce it at the very end. With the general formulas at hand, we use the Einstein-Cartan field equations to incorporate explicitly the spin of the matter. The resulting formulas fully describe the kinematics of dynamical spacetimes within the framework of the Einstein-Cartan gravity, while in the special case of the so-called Weyssenhoff fluid, they recover results previously reported in the literature. © 2017 American Physical Society.