A real estate investment trust (REIT) is a company running a funding pool that allows people to invest in real estate without physical purchase. Since REITs are stock market-traded real estate assets, there is debate as to whether their returns are driven by stock market risk factors. In this regard, this paper examines the impact of the well-established equity market risk factors of momentum, skewness, and kurtosis on the returns of different types of REITs, including mortgage REITs (MREIT), equity REITs (EREIT), and hybrid REITs (HREIT), across five countries—Australia, the UK, the US, Japan, and Canada—during the period 2000–2022, controlling for other well-established factors in the asset pricing literature. The study first adds the skewness and kurtosis to analyze cross-national REIT returns via the Fama–French five-factor model. Next, the cross-national REIT dataset is built for the different periods and then tested for the robustness of the effect of the factors during the COVID-19 period. Findings indicate that the influence of momentum on the return of the REITs is consistently positive across countries and different types of REITs. However, the significance of momentum for different REITs in different countries varies. These results were robust during the COVID-19 period, providing further confirmation that REITs behave less like stocks rather than real estate investments, with significant implications for investors.