"begin moving" is an acceptable phrase in written English.
You can use it when you want to indicate that something should be put in motion. For example, "Alice began moving the furniture to rearrange the living room."
He would suddenly begin moving furniture-tables, chairs, lamps.
"Slowly begin moving the raisin towards your lip.
Tenants will begin moving into the building in two weeks.
Morgan Stanley is expected to begin moving in next November.
His 10-person staff will begin moving in next week.
Harvey will begin moving potted trees and shrubs in plant nurseries.
But it will be two years before most existing claimants begin moving to PIP.
Triumphant Republicans will begin moving past the euphoria of midterm election celebrations.
"Ants that sense these chemicals stop moving, swing their antennae and then begin moving quickly.
However, Russian defence experts said troops would need at least 24 hours to begin moving.
And he said he hoped Apple could begin moving into the new building in 2015.
