Dualblade has the right of it, but to add to it - Shantotto has knowledge of spells that allow a person to travel across dimensions, she specifically states this during the original FFXI crossover event, and it was the spell that she mucked up in FFXI which started her add on pack storyline. Lightning ended up in Hydaelyn because the god of the FFXIII mythos Bhunivelze sent her there specifically to train her as preparation to become 'the saviour' in Lightning Returns ~ FFXIII, again this is clearly stated during those events. And Gilgamesh is a traveller of the Void who shows up in other FF worlds by mastering how to travel it (no thanks to X-Death banishing him there at the end of FFV).

Really, I think you're looking for something that isn't there Pinkie Pie. FFXI is not connected directly with FFXIV. They are separate worlds with separate storylines.