The outbreak of COVID-19 highlights the need for a more harmonized, less privacy-concerning, easily accessible approach to monitoring the human mobility that has been proved to be associated with the viral transmission. In this study, we analyzed 587 million tweets worldwide to see how global collaborative efforts in reducing human mobility are reflected from the user-generated information at the global, country, and the U.S. state scale. Considering the multifaceted nature of mobility, we propose two types of distance: the single-day distance and the cross-day distance. To quantify the responsiveness in certain geographical regions, we further propose a mobility-based responsive index (MRI) that captures the overall degree of mobility changes within a time window. The results suggest that mobility patterns obtained from Twitter data are amendable to quantitatively reflect the mobility dynamics. Globally, the proposed two distances had greatly deviated from their baselines after March 11, 2020, when WHO declared COVID-19 as a pandemic. The considerably less periodicity after the declaration suggests that the protection measures have obviously affected people's travel routines. The country scale comparisons reveal the discrepancies in responsiveness, evidenced by the contrasting mobility patterns in different epidemic phases. We find that the triggers of mobility changes correspond well with the national announcements of mitigation measures. In the U.S., the influence of the COVID-19 pandemic on mobility is distinct. However, the impacts varied substantially among states. The strong mobility recovering momentum is further fueled by the Black Lives Matter protests, potentially fostering the second wave of infections in the U.S.