Emigration mostly. This graph shows how many people emigrated to the United States in the nineteenth century. That constant removal of 2-3% of the population (in addition to movement to other countries) kept the population steady or declining.

In the late nineteenth and early twentieth century, the Irish had very limited economic opportunities. The countryside had about as many farmers as it could support, and there wasn't as prevalent urbanization as in the rest of the Anglophone world. Therefore, if you were a peasant of any means, it made a lot of sense to move elsewhere - particularly since there were bustling expat Irish communities in other countries.

Edit: Emigration is indeed what the Irish did, becoming immigrants.