Switched positive Persidskii systems with distributed and unbounded delays are studied. Right-hand sides of these systems are linear combinations of nonlinearities of a sector type. Special constructions of diagonal Lyapunov–Krasovskii functionals are proposed and conditions are derived under which the absolute stability of the considered systems can be proved with the aid of such functionals. The developed approaches are applied to the stability analysis of a mechanical system with switched nonlinear positional forces and to a problem of mobile agent deployment. Results of numerical simulations are presented confirming the theoretical conclusions.