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The phrase "move back to" is correct and can be used in written English.
It is typically used when referring to relocating to a place one had formerly lived in. For example, "After living abroad for five years, I decided to move back to my hometown."
Exact(60)
You move back to Boston.
"I can't move back to England.
She yearned to move back to China.
Some will move back to Iran.
Move back to New York?
Lettuce picking jobs move back to Mexico.
He decided to move back to New York.
Is a move back to Blighty imminent?
Or move back to Puerto Rico'.
I'm going to move back to Delaware".
They'll move back to the urban core.
More suggestions(18)
evolution back to
handover back to
move backward to
displacement back to
transformation back to
mobility back to
move backwards to
relocation back to
scroll back to
progression back to
keep coming back to
moving back to
move your feet back to
being transferred back to
being transported back to
is changed back to
move restored to
have evolved back to
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com