Many cities around the world have introduced dockless micromobility services and witnessed their rapid growth. Shared dockless e-scooters can expand mobility options for underserved neighborhoods and population groups, but they may also exacerbate existing transportation inequalities. So far, our understanding of the equity implications of dockless e-scooter services is limited. To address this research gap, this study develops an analytical framework to compare the equity performance of dockless e-scooter and docked bikeshare systems in Washington DC with a set of measures including availability, accessibility, usage, and idle time. Results reveal that dockless e-scooter services increase accessibility to shared micromobility options in disadvantaged neighborhoods but widen the access gaps across neighborhoods. Compared to bikeshare, shared e-scooters provide a higher level of spatial accessibility overall due to greater supply; however, the greater supply of shared e-scooters largely leads to longer average idle time rather than a greater number of trips. Finally, it appears that the bikeshare system's equity program is more effective than the equity programs provided by e-scooter operators in promoting usage in low-income neighborhoods. These findings suggest that increasing vehicle supply alone would probably not lead to higher micromobility use in disadvantaged neighborhoods. Instead, policymakers should combine a variety of strategies such as promoting the enrollment of equity programs and reducing access barriers (e.g., smartphone and banking requirements) to micromobility services.