Time traveling and a certain pair of goggles.

[Riastrad]

My thinking is, if the Echo was capable of time travel, Emet Selch wouldn't have been so desperate to extract the secrets of time travel (as well as the future following the Calamity of Light) from the Exarch.

I feel there is something far more to the Echo than just going on vision quests at inopportune times. As Lahabrea once said, "Very well. Let us show these mortals the true power of the Echo. The power to break down the barriers of existence!". So much is still not understood as to the nature of our echo. Krile's allows her to see traces of souls if using a strong enough foci, while Fordola's own artificial Echo allows her to see minor shifts in aether, effectively giving her tip off when her opponent is going to attack. The Sahagin Priest's echo in patch 2.2 allowed him to move his soul between bodies after he was killed by Merlwyb.

Ours seems to (from a gameplay standpoint at least) allow us reattempt fights after we have been defeated by allowing us to slip back into an earlier moment in time before the battle. Giving us (the player) an idea of what to expect as we have already tried and failed to do. It's also shown in 1.0 (even if it was retconned, and no I don't buy that in the slightest as that is the easy way out that creates even more problems than if it was left in) that it is possible for another echo user to discern if someone is present in a past event they were not part of by using their own echo. It's how we joined the Path of the Twelve in 1.0, and that is how Minifila recognized us in ARR when no one else did.

I suppose at this point i'm just fueling my own head cannon in an attempt to explain away inconsistencies (dreadful I know). But it also includes a way for ideas and information to be passed back in time. Much like Squall does in FFVIII. (again a big influence on the current raid)