We address the issue of what happens when the unitary evolution of a generic closed quantum system is interrupted at random times with non-unitary evolution due to interactions with either the external environment or a measuring apparatus. We adduce a general theoretical framework to obtain the average density operator of the system at any time during the dynamical evolution, which is applicable to any form of non-unitary interaction. We provide two explicit applications of the formalism in the context of the so-called tight-binding model relevant in various contexts in solid-state physics, for two representative forms of interactions: (i) stochastic resets, whereby the density operator is at random times reset to its initial form, and (ii) projective measurements at random times. For (i), we demonstrate with our exact results how the particle is localized on the sites at long times, leading to a time-independent mean-squared displacement of the particle about its initial location. For (ii), we show that repeated projection to the initial state of the particle results in an effective suppression of the temporal decay in the probability of the particle to be found on the initial state. The amount of suppression is comparable to the one in conventional Zeno effect scenarios, but which does not require to perform measurements at exactly regular intervals that are hallmarks of such scenarios.