In this paper, we consider a wide class of time-varying multivariate causal processes that nests many classical and new examples as special cases. We first show the existence of a weakly dependent stationary approximation to initiate our theoretical investigation. We then consider a quasi-maximum likelihood estimation (QMLE), and provide both point-wise and uniform inferences to coefficient functions of interest. The theoretical findings are further examined through extensive simulations. Finally, we show empirical relevance of our study by evaluating both temporal and contemporaneous connectedness between the stock markets of China and U.S.