We prove a central limit theorem for families $\{X_n(N): \mathbf{n} \in \Lambda(N)\}$ of associated random variables indexed by subsets $\Lambda(N)$ of $\mathbb{Z}^d$, as $N \rightarrow \infty$; this is an extension of the Newman-Wright invariance principle for associated stationary sequences $\{X_n: n \geq 1\}$ satisfying $\sum_n \operatorname{cov}(X_1, X_n) < \infty$, but with the stationarity property replaced by conditions on the moments of the $X$'s. The theorem has applications to the voter model and the percolation model. In the latter case, it provides an extension of a central limit theorem of the authors [4], by reducing the severity of the moment conditions. Also, we prove a central limit theorem for certain non-stationary non-associated families of random variables which arise in percolation theory. This includes, for example, a central limit theorem for the number of open clusters contained within the circuit $\gamma(n)$ of $\mathbb{Z}^2$, where $\{\gamma(n)\}$ is a sequence of circuits which satisfy a regularity condition and whose interiors $\{\overset{\circ}\gamma(n)\}$ satisfy $|\overset{\circ}\gamma(n)| \rightarrow \infty$ as $n \rightarrow \infty$.