We propose a straightforward algorithm to estimate large Bayesian time-varying parameter vector autoregressions with mixture innovation components for each coefficient in the system. The computational burden becomes manageable by approximating the mixture indicators driving the time-variation in the coefficients with a latent threshold process that depends on the absolute size of the shocks. Two applications illustrate the merits of our approach. First, we forecast the US term structure of interest rates and demonstrate forecast gains relative to benchmark models. Second, we apply our approach to US macroeconomic data and find significant evidence for time-varying effects of a monetary policy tightening.