This article extends weak convergence bounds of Markov transition kernels to convergence bounds on the variance of the Markov kernel applied to Lipschitz functions. In the reversible case, weak convergence rates of the transition kernels imply chi-squared divergence convergence bounds if the density of the initialization measure is Lipschitz. These results provide new tools to establish central limit theorems for Lipschitz functions used in Markov chain Monte Carlo simulations. Applications are explored to the stability of Metropolis-Hastings algorithms in high dimensions, stochastic gradient descent, and solutions to stochastic delay equations.