"fall behind" is correct and usable in written English. To use it, you just need to make sure it is used in the correct tense. For example, "She fell behind on her rent payments and now owes her landlord two months worth of rent.".
You don't fall behind".
So why did we fall behind?
"If you don't change, you fall behind".
When did it begin to fall behind?
He was going to make sure I didn't fall behind".
The Pacific Time Zone will fall behind by another hour.
"I don't want him to fall behind like I did".
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