Time traveling and a certain pair of goggles.

[Enkidoh]

The Echo is not time travel though at all (that instance with the pistol in the 1.0 Limsan storyline was merely a plot hole that SE appears to have since quietly ignored for the sake of consistancy), it's simply accessing and changing another's memories of an event. Us appearing in an Echo vision of when Cid defected to Eorzea to give Cid his goggles was merely a visual metaphor for us restoring his lost memories, it was a symbol of his past, and wasn't an actual event that really took place (because the Echo can also alter a person's memories to reflect that).

This is after all why SE also changed how the Echo was depicted in ARR from how it was in 1.0 where an Echo user could no longer "use it at will" but would instead randomly and uncontrollaby activate, as how it was originally shown in 1.0 where the player's character learned how to control it and use it at will (under Minfilia's urging no less!), meant it had... disturbing implications, which SE are now probably trying to get away from.