Over the last decade, the lives of children in several countries, including Greece, have been affected by recession. The aim of the present study was (a) to examine time trends in physical activity (PA) and screen time (ST) of Greek preschool children, together with their family affluence (FA), from 2009 until 2018, and to explore the associations among them; and (b) to investigate the connections of parental educational level and children’s BMIs to their achieving ST (<1 h/day) and PA (11,500 steps/day) guidelines. A total of 652 children from four cross-sectional cohorts participated. PA was recorded with Omron HJ-720IT-E2 pedometers, whereas ST, family affluence (FA) and parental educational level were reported by participants’ parents. The results of the one-way ANOVAs that were computed revealed statistically significant differences among cohorts, albeit of no practical importance, in PA, ST and FA. According to the regressions calculated, neither BMI nor the educational level was related to membership in ST and PA guidelines groups. ST was a significant predictor of children’s PA in all week periods (school-time, leisure-time, weekend), whereas FA was not such a strong predictor. Multilevel interventions aiming at both ST and PA seem to be imperative for the benefit of young children’s health.