“What are you, part pack mule? Though those favoring heavier weapons and armor are likely to take this perk on general principle, it's also hugely beneficial to scavengers; spend more time in the wilderness gathering loot to sell or trade! Carry a greater complement of killing equipment! And at higher levels, have some way of easily maneuvering back to safety without dropping your ill-gotten gains.”— Fallout 4 Vault Dweller's Survival Guide description
With rank 3 of this perk, running while overencumbered consumes AP similarly to sprinting. Because of this, increased Endurance also lowers the AP drain just like it would for sprinting.
Rank 3 of the Moving Target perk also takes effect with the ability granted by rank 3 of this perk, reducing the AP drain of running while overencumbered. It also combines with rank 5 of this perk multiplicatively, further reducing (but not negating) the AP drain.
On Survival Mode, Rank 4 will negate damage taken from being overencumbered.
A bug with rank 3 will use your Fusion Core energy, Since it still drains AP. This will quickly use up any Fusion Core energy from your suit but only when moving. as mentioned above Moving Target and Rank 5 of Strong Back will reduce the AP drain. But you can never negate AP drain on moving thus will still continually drain your Cores while moving.