There is not much known about this version of Kiritsugu. In Fate/kaleid liner PRISMA☆ILLYA, he is alive and is happily married to Irisviel von Einzbern though he is often away overseas; unlike his counterpart, he apparently chose to prevent the 4th Holy Grail War which later led the Einzberns to ruin as Irisviel mentioned that it no longer exists. He adopted Shirou at a young age. Shirou wishes Kiritsugu to come back home as the female to male ratio is increasing.
From what little we see of him during flashbacks as well as the 2nd anime opening, he looks very much like his original Fate/Zero counterpart. However, due to the angles at which he's shown, his eyes are hidden from sight.
From what can be gathered in his very brief screentime, he's shown to genuinely care for both Iri and Illya despite his initially cold-looking personality. He cares for them so much, that he elects to abandon his ideals, and by extension his ties with the Einzberns and the Holy Grail War altogether, starting a new life for his daughter in Fuyuki instead. This is not different to his Fate/Zero counterpart, who wished to run away with his family but could not due to Irisviel's decision to sacrifice herself for the Grail to make a better world for Illya.
Fate/kaleid liner PRISMA☆ILLYA Edit
During the events of Fate/kaleid liner PRISMA☆ILLYA, Kiritsugu only makes a few appearances in Illya and Kuro's flashbacks. His only direct appearance was when he was seen talking with Irisviel overseas in Fate/kaleid liner PRISMA☆ILLYA 2wei!. While he was smoking, Irisviel got lost in her thoughts, so Kiritsugu asked her what was wrong. She then responded by saying that she was under the impression Illya got into another troublesome situation on the other side of the ocean. He tells her that two weeks ago, she said that same thing and went home to Japan, asking her if her bond with Illya doesn't diminish over distance. She finishes the conversation by admitting that he is right and that it's her mother's intuition.
In Fate/kaleid liner PRISMA☆ILLYA 3rei!!, Kuro thinks about him while mentioning to Angelica Ainsworth that she knows someone like them because of their ideology of saving the many over the few.