This paper explores the transmission of risk from the United States equity market to the equity markets of the BRICS countries using a multivariate quantile process. The focus is on the contagion effect at the extreme quantiles, both upside and downside. In addition, a pseudo-impulse-response function (PIRF) analysis is conducted to investigate the responses of the five emerging stock markets to a shock in the US market. We conduct an empirical study against the backdrop of the COVID-19 event and the Russia-Ukraine conflict, finding that risk spillovers between the US stock market and the five emerging stock markets are significantly enhanced during the COVID-19 period. Moreover, a shock in the US market produces a stronger and more persistent negative effect at the downside quantiles compared to upside quantiles. However, we find little evidence of cross-market risk spillovers among the investigated variables during the Russia-Ukraine conflict period. We also discuss the implications of these findings for investors and policymakers in terms of portfolio holdings and policy coordination.