The method of nonlocal reduction has been proposed by G.A. Leonov in the 1980s for stability analysis of nonlinear feedback systems. The method combines the comparison principle with Lyapunov techniques. A feedback system is investigated via its reduction to a simpler "comparison" system, whose dynamics can be studied efficiently. The trajectories of the comparison system are explicitly used in the design of Lyapunov functions. Leonov's method proves to be an efficient tool for analysis of Lur'e-type systems with periodic nonlinearities and infinite sets of equilibria. In this paper, we further refine the nonlocal reduction method for periodic systems and obtain new sufficient frequency-algebraic conditions ensuring the convergence of every solution to some equilibrium point (gradient-like behavior).