We develop a design-based framework for causal inference that accommodates random potential outcomes without introducing outcome models, thereby extending the classical Neyman--Rubin paradigm in which outcomes are treated as fixed. By modelling potential outcomes as random functions driven by a latent stochastic environment, causal estimands are defined as expectations over this mechanism rather than as functionals of a single realised potential-outcome schedule. We show that under local dependence, cross-sectional averaging exhibits an ergodic property that links a single realised experiment to the underlying stochastic mechanism, providing a fundamental justification for using classical design-based statistics to conduct inference on expectation-based causal estimands. We establish consistency, asymptotic normality, and feasible variance estimation for aggregate estimators under general dependency graphs. Our results clarify the conditions under which design-based inference extends beyond realised potential-outcome schedules and remains valid for mechanism-level causal targets.