"Casting will always be interuppted [sic] upon receiving direct damage."

[Aegis]

Gap Closers:

MNK: Shoulder Tackle, Fists of Wind (not technically a gap closer per se)
DRG: Jump/Spineshatter/Dragonfire dive all three Dragoon jumps will place them next to the target (at least that's how I interpret the 'adjustment' they are making to jump's positioning).
WAR: Holmgang

Pushbacks/distancers

BLM: Aetherial Manipulation
WHM: Fluid Aura
BRD: (Forgot the name of it, evasive shot, something like that?)
DRG: Evasive Jump
SMN: Garuda has a pushback move

I think Scholar and Paladin are the only jobs without some form of distance manipulation.

[Kiokin]

Gap Closers:

MNK: Shoulder Tackle, Fists of Wind (not technically a gap closer per se)
DRG: Jump/Spineshatter/Dragonfire dive all three Dragoon jumps will place them next to the target (at least that's how I interpret the 'adjustment' they are making to jump's positioning).
WAR: Holmgang

Pushbacks/distancers

BLM: Aetherial Manipulation
WHM: Fluid Aura
BRD: (Forgot the name of it, evasive shot, something like that?)
DRG: Evasive Jump
SMN: Garuda has a pushback move

I think Scholar and Paladin are the only jobs without some form of distance manipulation.

You're interpreting it wrong. All they're doing is making it so DRG have control of their character faster out of a jump.