In binocular rivalry, the visual percept alternates stochastically between two dichoptically presented stimuli. It is established that both processes related to the eye of origin and binocular, stimulus-related processes account for these fluctuations in conscious perception. Here we studied how their relative contributions vary over time. We applied brief disruptions to rivalry displays, concurrent with an optional eye swap, at varying time intervals after one stimulus became visible (dominant). We found that early in a dominance phase the dominant eye determined the percept by stabilizing its own contribution (regardless of the stimulus), with an additional yet weaker stabilizing contribution of the stimulus (regardless of the eye). Their stabilizing contributions declined in parallel with time so that late in a dominance phase the stimulus (and in some cases also the eye-based) contribution turned negative, favoring a perceptual (or ocular) switch. Our findings show that depending on the time, first processes related to the eye of origin and then those related to the stimulus can have a greater net influence on the stability of the conscious percept. Their co-varying change may be due to feedback from image-to eye-of-origin representations