follow through with/on doing sth

[AlexAD]

Hello.

I definitely heard, 'He didn't follow through on writing the book' but the entry from the OALD has only this, follow through with doing sth.
Are they both correct? Any guess why the OALD is missing 'follow through on'?

Thanks, Alex.

[emsr2d2]

Hello.

I definitely heard, 'He didn't follow throw on writing the book' but the entry from the OALD has only this, follow through with doing sth.
Are the both correct? Any guess why the OALD is missing 'follow through on'?

Thanks, Alex.

I think that the norm is to use "follow through with" or "follow up on". They don't mean the same thing, of course.

He followed through with his promise to write a book about his life.
He followed up on the information he had received the previous day.