Starting from certain causality conditions and a discrete form of general covariance, we derive a very general family of classically stochastic, sequential growth dynamics for causal sets. The resulting theories provide a relatively accessible "half way house" to full quantum gravity that possibly contains the latter's classical limit (general relativity). Because they can be expressed in terms of state models for an assembly of Ising spins living on the relations of the causal set, these theories also illustrate how non-gravitational matter can arise dynamically from the causal set without having to be built in at the fundamental level.