"some pushing" is correct and usable in written English.
You can use this phrase to describe an action that is gentle or for a minimal effort. For example, "The door was stuck, but with some pushing it opened."
Watergate needed some pushing to get started.
There was some pushing and shoving, women wailed.
There was some pushing and shoving, but no serious skirmishes.
"There have been some pushing and shoving incidents".
But outside, he said, "there was some pushing" among a crowd of youngsters.
It's swung in, but some pushing in the box leads to a free-out for Marseille.
With some pushing and tugging and lifting, the rooftop crowd soon set up the windmill.
They were mostly families, some with their dogs, some pushing strollers.
They said the would-be robbers made off empty-handed after "some pushing and shoving".
The teams were called for offsetting penalties for some pushing and shoving.
It captures a long row of women, some pushing prams, entering what looks like a somewhat austere rest home.
