
Three men standing on deck of the Liguria, en route to Australia, 1951. Photograph held in the State Library of Victoria in Melbourne, Australia. Public domain.
Some self-appointed grammarians appear to have created a ‘rule’ that is confusing confused people even further. The ‘rule’ — I have been told twice in the last month — is that the verb emigrate always goes with the preposition from, whereas the verb immigrate always goes with the preposition to. As if these are verb–preposition collocations … They are not.
I don’t know where this ‘rule’ comes from. I had never heard of it before. After a brief search online I found this clear, black-and-white explanation that made me scratch my head:

Source: https://www.grammarbook.com/homonyms/emigrate-immigrate.asp (accessed 4 November 2017)
First, the author gives two correct examples that comfortably use both from and to with either verb, blithely contradicting the ‘rule’ they’re supposed to illustrate.
He emigrated from Russia to America.
He immigrated to America from Russia.
Then, the author gives two examples that are ‘incorrect’. These sentences are copies of the correct examples, except for some information that is left out …
He emigrated from Russia to America.
He immigrated to America from Russia.
Leaving out information does not necessarily make these sentences incorrect. We leave out information all the time when we deem it to be understood (because it’s been mentioned before, or because it is general knowledge) — it is called ellipsis. So, assuming the reader knows Isaac Asimov is an American author who was born in Russia, either of the following sentences is perfectly correct.
Isaac Asimov’s family emigrated to America when he was three years old.
Isaac Asimov’s family immigrated from Russia when he was three years old.

Mrs. Czeslawa Galaska and a friend, near the Bonegilla Migrant Reception and Training Centre, 1950. Photograph held in the State Library of Victoria in Melbourne, Australia. Public domain.
Like come and go, emigrate and immigrate are verbs of motion and involve a source and a goal, either explicitly or implicitly, which is why we can use either from and to (or both, or neither) with these verbs. If you know that the Asimov family was Russian, it is understood that their migration would have started in Russia. If you know that Isaac Asimov was an American author, it is understood that the family’s migration ended in the USA.
So, there is no ‘rule’ about which prepositions these verbs take …
What can be confusing is whether to use the verb emigrate or the verb immigrate. This depends on whether the focus of the text in which the sentence sits is on the starting point or source of the migration, in which case you would use emigrate (e- from the Latin ex: out of, from), or on the endpoint or goal of the migration, in which case you would use immigrate (im– from the Latin in: into). However, sometimes that focus is not that clear-cut …
Of course, you could also use migrate, or move!