Of the main points of evidence for an Italian migration is this from Eleazar of Worms who was born in 1176 in Germany
“They all received the secret of the true version of the prayers, teacher from his teacher, up
to Abu Aaron, the son of Rabbi Samuel ha-Nasi, who came from Babylonia because of a
misadventure, and had to wander from place to place [as a punishment], until he came to
the country of Lombardy, to a city named Lucca, where he found Rabbi Moses [son of
Kalonymus], who wrote the piyyut, Eimat Norotekha, and he [Abu Aaron] transmitted to
him all his secrets. And he was Rabbi Moses, son of Kalonymus, son of Rabbi Judah. He was
the first who left Lombardy, he and his sons, Rabbi Kalonymus and Rabbi Jekuthiel, and his
relation Rabbi Ithiel, and other important persons; for the king Charles brought them with him
from the country of Lombardy, and settled them in Mainz, and there they multiplied and
flourished very much; until God's fury hit all the holy communities in the year 1096. And then
we were all lost, all perished, except very few who were left from our kinsmen. [Rabbi
Kalonymus] the Elder who transmitted [the Torah] to Rabbi Eleazar he-Ḥazzan of Speyer,
as we have written above, and Rabbi Eleazar he-Ḥazzan transmitted it to Rabbi Judah he-
Ḥasid, and from him I, the small one, received the secrets of the prayers and other secrets.”
He describes the migration of the Kalonymus family from the Kingdom of Lombardy and also states that other important persons were brought to so this may imply a small migration depending on what he mean by others it could be a few dozen or a few hundred
