This paper investigates the limiting behaviour of degree-degree correlation metrics for sequences of random graphs under a general assumption of local convergence in probability. We establish convergence results for Pearson's correlation coefficient r, Spearman's rho, Kendall's tau, average nearest neighbour degree (ANND), and average nearest neighbour rank (ANNR). Our results explicitly show how the limits of these degree-degree correlation metrics depend on the local structure of the graph. We then apply our general results to study degree-degree correlations in rank-1 inhomogeneous random graphs and random geometric graphs, deriving explicit expressions for ANND in both models and for Pearson's correlation coefficient in the latter one. Keywords: random graphs, degree-degree metrics, neutral mixing