Their coming was foretold by The Time Keepers during The Destiny War, Genis-Vell mentions to the present day Wasp after she asks him about The Infinities that they would appear in her future very soon. Several months later a Rigellian colony and The Solar System it inhabited was annihilated by a group of robotic entities called The Servitors. Thor, having heard of this atrocity and about those whom the Servitors served, led a team of "Cosmic" Avengers to the Solar System where the Servitors laying waste to other nearby inhabited worlds. After a brief fight several Celestial-like entities called Walkers appeared and began building a massive ring shaped device after destroying several of the planets. The Avengers immediately mistook The Walkers for The Infinities but when a hand larger than the entire Galaxy itself emerged from The Dimension of Manifestations and activated the ring, which in turn linked all of the Stars in that Galaxy together, The Avengers knew their foes were way beyond their league.
Remembering the time he had accidentally summoned Eternity instead of an Elder of the Universe like he had planned, Quasar suggested that Moondragon link The Avengers' minds together so that they could reach out to Eternity and call for his help. The summoning worked and Eternity forced the hand off of the ring but ended up being pulled by it into The Dimension of Manifestations. Upon arrival The Avengers found themselves inside Eternity who was now being confronted by The Infinities themselves. Almost immediately The Infinities demanded to know who he was and why he was interfering with their work. Eternity explained that their actions had cost the lives of countless people. The Infinities were initially perplexed upon learning of the existence of Mortals, the very concept of there being life that tiny in comparison to The Infinities seemed to be absurd at first but after speaking with The Avengers (who used Eternity as a conduit through which they could communicate) The Infinities were horrified by their actions.
All this time they had been realigning the energies of each Universe they traveled to in order to make the flow more efficient but because their Galaxy-Grappling Rings only latched onto and protected the Stars, any inhabited world being pulled along with their Galaxy to a point where The Infinities were building a Wall of Galaxies would be torn apart by the unimaginable tidal forces. From that day foreward The Infinities abandoned their "experiment" and began making amends. One of them even sacrificed his lifeforce to restore the Galaxies ruined by their actions on Earth-616 but was unable to bring those who had died at the hands of The Servitors back to life. However he did promise that life would eventually return to the revived worlds. As the Infinity died he stated that he was going to the afterlife to beg for forgiveness to those The Infinities had accidentally killed during their travels. The Infinities' current whereabouts are unknown.