Looking on the internet deeply has found these results:

move towards is the most popular phrase on the web.

go towards 18,050,000 results on the web	move towards 19,960,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
It is impossible to distinguish between the various elements that go towards making the Lisbon strategy so effective That money was supposed to go towards school, like I said. I saw you go towards the gate. From there we can go towards the Turkish border. Tell her to go towards the light. and I am talking here of its theoretical effectiveness, not the enthusiasm with which it is implemented. We're not supposed to go towards the noise. He had to go towards the water as the sirens sounded.	It doesn't look like this phrase is very popular! The search could not find any examples on the internet.

go towards

18,050,000 results on the web

move towards

19,960,000 results on the web

More popular!

It is impossible to distinguish between the various elements that go towards making the Lisbon strategy so effective
That money was supposed to go towards school, like I said.
I saw you go towards the gate.
From there we can go towards the Turkish border.
Tell her to go towards the light.
and I am talking here of its theoretical effectiveness, not the enthusiasm with which it is implemented.
We're not supposed to go towards the noise.
He had to go towards the water as the sirens sounded.

It doesn't look like this phrase is very popular! The search could not find any examples on the internet.