Installed at the request of animal rights activist, the more spacious cages allow the animals to move at their full natural speed.
Installed at the request of animal rights activist, the more spacious cages allow for the animals moving at their full natural speed.
Installed at the request of animal rights activist, the more spacious cages allow the animals to move at their full natural speed.
Installed at the request of animal rights activist, the more spacious cages allow for the animals moving at their full natural speed.
The idiom "allow A to do X" has the connotation of conscious design or intent. The idiom "allow for" would be used for an unintended consequence of a mechanical design, which is not appropriate here.