This study assesses whether long-term institutional investors help stabilise or destabilise stock markets, taking into account international spillovers arising from these investors' portfolio rebalancing activities. Based on granular data of individual insurance companies and pension funds and their exposures to global markets, our results show that these investors notably change their fund flow dynamics among international markets in times of financial stress, during which the fund flows to developed economies overall are increasingly correlated with those to all other economies. Meanwhile, these long-term institutional investors, who stabilise declines in global stock markets during normal market conditions, destabilise stock markets in major developed economies during adverse market conditions. In view of such different impacts on stock markets, investors and regulators, as well as policymakers, should be mindful of any potential outcomes of the investment behaviour of these institutional investors during different market conditions.