The effectiveness of policy measures in the midst of a public health crisis hinges upon widespread compliance with the ordinances. This study explores how leaders' discourse relates to compliance with stay-at-home orders to mitigate the spread of COVID-19 in Mexico. It uses county-level mobility data from geolocation events of smartphone users and daily briefings from the President and the coronavirus czar. Quantile regressions, sentiment analysis, and topic modeling are used to shed light on the dynamics between political discourse and containment of mobility. Discrepancies were found between the content and tone of the President's messages and those of the coronavirus czar. These contrasts translate into differences in the association between the content of the speeches and mobility patterns. While virtually no association was found between mobility and the President's discourse, mentions of the stay-at-home orders are associated with up to a five percentage point decrease in mobility. Mentions of other guidelines such as washing hands, wearing masks, and social distancing predict a 2-8 percentage point increase in mobility in the interquartile range of the mobility distribution. The discourse's association to mobility is mediated by awareness of COVID-19-related topics and individuals' political affiliation.