It's also worth noting that names weren't as fixed in the past as we think of them now. Just as English words didn't have consistent spellings until at least the 1600s and often later, names didn't either (meaning a son spelled his last name differently from how his father spelled it). It was also common to change your name completely after moving to a new country. One example is the many French immigrants to England during the Huguenot persecution of the 1500s that changed their French surnames into English translations of those names.

I am not as familiar with 19th and early 20th C immigration to the US, but I would imagine some of those factors were still true then as well (both creating new names and changing spelling of existing names, the latter certainly influenced by the double-digit illiteracy rates of foreign-born residents in the US up until at least the 1940s).

As for names changing in the actual process of immigration, I've worked recently with a number of refugees and seen the challenge of registering a name in the US that is written in a different character set (eg Arabic or Chinese). These names are recorded as a transliteration into the English alphabet, which is then the person's legal name in the US forever. So that definitely still happens quite a bit.