The phrase "allowed to move through" is correct and usable in written English.
It can be used in contexts where permission or capability to pass through a space or situation is being discussed.
Example: "In the event of an emergency, all personnel are allowed to move through the designated exits without delay."
Alternatives: "permitted to pass through" or "authorized to traverse".
Only ambulances, news media and diplomatic vehicles were allowed to move through the capital.
Regarding the list of tradable goods that are allowed to move through the border, Pakistan offered to move from a positive list to a negative one, which could open up trade by a substantial amount if Delhi accepts the proposal.
But like the water that sustains the Everglades, the water that is allowed to move through the Sacramento-San Joaquin River Delta and into the Pacific is not wasted.
In such an environment, each student is allowed to move through a curriculum at his or her own pace, and no one moves on until s/he can demonstrate mastery of the core concepts.
Even watching Larry try to do something good, like warn his friends about a possible terrorist attack, and watch it go hilariously wrong, made it feel like my own annoyances at the faux-niceties bred by social media or that coming with having to be a pleasant person allowed to move through the world were validated.
But regardless of what they discover, a wider debate has now begun about the wisdom of letting so many unknown migrants enter Europe through its southern borders, and being allowed to move onwards through the continent with few restrictions.
The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh.
Each mouse was placed at the end of one fixed arm and allowed to move freely through the maze during an 8-min session.
The mice were allowed to move freely through a circular tunnel between the two compartments.
The mouse is allowed to move freely through the maze for 8 minutes.
Each mouse was placed at the end of one arm, and allowed to move freely through the maze during an 8 min period.
