In modeling multivariate time series, it is important to allow time-varying smoothness in the mean and covariance process. In particular, there may be certain time intervals exhibiting rapid changes and others in which changes are slow. If such locally adaptive smoothness is not accounted for, one can obtain misleading inferences and predictions, with over-smoothing across erratic time intervals and under-smoothing across times exhibiting slow variation. This can lead to miscalibration of predictive intervals, which can be substantially too narrow or wide depending on the time. We propose a continuous multivariate stochastic process for time series having locally varying smoothness in both the mean and covariance matrix. This process is constructed utilizing latent dictionary functions in time, which are given nested Gaussian process priors and linearly related to the observed data through a sparse mapping. Using a differential equation representation, we bypass usual computational bottlenecks in obtaining MCMC and online algorithms for approximate Bayesian inference. The performance is assessed in simulations and illustrated in a financial application.