[servicepub (RIP)]

Understand that Regimental Numbers were issued at point of enlistment. In the early days of the war enlistment and training were a unit's responsibility. It was soon realised that this was inefficient so recruiting centres were established as well as training centres.
This meant that early volunteers signed up at the unit and usually stayed with the unit. Later recruits were from the area/district and would be posted to a unit after they finished their training (at various levels of training centres).
Typically, soldiers were sent to a reinforcement centre in the UK for continued and specialist training and from there sent to a unit that had lost troops in action. This is how a soldier from BC could end up in a NB unit.