If the USA and Mexico set different thresholds for the poverty line, with the Mexican threshold being lower, then people above the Mexican poverty line but below the USA poverty line could emigrate. This would raise the poverty rate in both countries, but this is an artifact of the two countries differing definitions of poverty. But if you measured poverty in the entire USA+Mexico region using a single definition of poverty, you would see it decrease, assuming those immigrants are better off after moving. (Note that actually measuring poverty this way is difficult due to factors like differing currencies with exchange rates varying over time, so the concept of a unified poverty definition is mostly just theoretical.)

Fun fact: most immigrants from Mexico are not poor-relative-to-Mexico.