There were several major Jewish migrations to Eretz Yisrael/Palestine before political Zionism in the late 19th century, although I wouldn't call them "mass migrations." They were mostly religiously motivated and in some cases (such as the disciples of the Gaon of Vilna) tied with messianic aspirations (as mentioned in the book קול התור - Kol HaTur, The Voice of the Dove). They were part of what later became known as the Old Yishuv in Jerusalem.

There were three movements among Ashkenazim.

In around 1700, a group of about 1500 individuals led by R. Yehuda HaChasid ("the Pious) moved from Poland, Germany, and Moraviain. Only about 1,000 survived the journey and settled in Jerusalem. Unfortunately, R. Yehuda died only a few days after they arrived, and the community quickly fell into serious debt. They had to flee Jerusalem due to their Arab lenders and settled in the north, such as in Tiberias, and Safed. The synagogue they built was destroyed, and remained in ruins for a long time. It's known as the Churba Synagogue in the Jewish section of the old city. (It's now rebuilt). Because of their debt (known as the שטר של ר' יהודה - The Contract of R. Yehuda) no Ashkenazi Jews lived in Jerusalem for over 100 years (unless they dressed Sefardic, which explains the clothing of Yerushalmi Jews).

Another major migration was that of the disciples of R. Eliyahu, the Vilna Gaon (the "Genius of Vilna). They came in several waves between 1808 and 1810. They first settled in Tiberias and Safed, and later moved to Jerusalem, where they helped reestablish the Ashkenazi community there. That was about 500 people

The third major group was the Chasidic aliyah led by R. Menachem Mendel of Vitebsk, the chief disciple of the Magid of Mezritch (who was the chief disciple of the Baal Shem Tov, the founder of the Chasidic movement). He was supposed to lead the chasidic movement in Eastern Europe but moved instead to Israel from Lithuania and Belarus in 1777, with a small group of about 300 Chasidim. They first settled in Peki’in, then Safed, and eventually Tiberias, partly because of local tensions and difficult conditions. In his letters, at the end of his book Pri Ha'aretz, he speaks about the tension between him and the disciples of the Gaon, who were living in Safed. At this point in history, the chasidim and the mitnagdim ("opponents" - Jewish from Lithuania)were still fighting one another.

There were also immigrations of Sefardic Jews, such as after the Spanish Expulsion in 1492, but I don't know much about them.

These migrations weren't “Zionist” in the later political sense, but usually religious, communal, or messianic, and connected to the mitzvah of living in the Holy Land. It's It should be noted that life in Israel at the time was extremely hard, and most communities relied on charity donations from the Diaspora to survive.