When a single equipment grounding conductor is run with multiple branch circuits in the same raceway or cable, how shall it be sized?

This will be according to the short circuit calculation.
The maximum allowable current and the duration of fault will be considered.
However, the allowable duration is usually one second.
Actually, in low voltage system, the practice is not to lay a separate cable or wire for grounding.
Instead, the shielded copper of the low voltage cable will be heat-shrinked, lug-pressed and earth the equipment.

It shall be sized for the largest overcurrent device protecting conductors in the raceway or cable.

this will be according to the short circuit calculation.
The maximum allowable current and the duration of fault will be considered.

this will be according to the short circuit calculation.
The maximum allowable current and the duration of fault will be considered