This work contributes to the discussion on how innovative data can support a fast crisis response. We focus on the use of two different sources of operational data from Meta to gain useful insights on migration flows following the conflict in Ukraine. By analysing the Facebook Ukrainian speaking Monthly Active Users, we estimate the flows of people fleeing the country towards the European Union. Our results indicate (at the fifth week since the beginning of the war), an increase in Ukrainian stocks in all the European Union countries, with Poland registering the highest percentage share (33%) of the overall increase, followed by Germany (17%), and the Czech Republic (15%). We then check the correlation of the Facebook Social Connectedness Index with the Ukrainian diaspora at regional (NUTS-3) level. Our analysis suggests that the index is a good proxy of the Ukrainian diaspora at regional scale, although its use to estimate the actual volumes still needs further study. In conclusion, our work shows how innovative data from social media could provide initial insights on the conflict-induced international flows in almost a real-time, and at sub-national resolution, potentially supporting initiatives aimed at providing humanitarian assistance to the displaced people.