Can you think of a book/comic/movie/tv show about someone who goes to look for a baby or child that they believe is their dead loved one reincarnated?
I asked about a vaguely similar story a few years ago and people realized I was thinking of was Made In Heaven.
It kind of fits your inquiry.
Not babies, but…
The Hawkman/Hawkwoman story in The Flash and Legends of Tomorrow, and probably the comics as well. The idea is that they’re a pair of ancient lovers who are destined to die and be reborn and find each other and repeat this cycle eternally.
Didn’t go looking for him, but in the Magic’s Price trilogy of novels, Vanyel’s lover Tylendel commits suicide and then is reborn as Stefan, and they eventually reunite (and figure out the whole reincarnation thing because magic).
oh… were you looking for a specific story, or just suggestions about the theme?
Audrey Rose. The movie version starred Anthony Hopkins, IIRC. (Oh, and I second the Herald Mage trilogy. EXCELLENT series)
Just chiming in to third the series rec 
Suggestions about the theme.
Would not looking for a loved one, but finding one while looking for the reasons for amnesia/nightmares do? That’s Dead Again. The blurb at IMDB reads,
There is suspense and mystery and dread.
James Halperin’s The First Immortal is sort of about that kind of thing, sort of.
Spider Robinson had a short story titled “Soul Search” in which it had been discovered that reincarnation was a fact. A wealthy woman attempted to bring back her dead husband but got instead the spirit of someone else. It was also discovered that when a person dies and their spirit goes to another body, it does not retain the memories it once had. So her quest was for naught. But then another man who loves her…well, one can find and read the story for yourself. I think it’s pretty good. And my copy of the book it’s in is sgned by the author!
Katherine Kerr’s Daggerspell series. But in that case, before he can find her again as a lover he has to not only repay the wrong he initially did her which led to her death, but also essentially give up the possibility of any sort of romantic relationship … and then get it after all in the end.
It takes a lot of books.
Vectors by Michael Kube-McDowall plays in that sandbox too, but in a slightly different manner.
This motif and related devices show up in a number of Bollywood movies, unsurprisingly since reincarnation is a chief tenet of Hindu belief.
“The Reincarnation of Peter Proud” isn’t quite what you mention…but still…
Not exactly what you are looking for, but an interesting read
http://www.goodreads.com/book/show/664205.The_Search_for_Grace
I recommend the movie I Origins.
So…are all of these suggestions just sort of similar, or do any of them involve a person seeking out or stumbling upon a child who was their parent, sibling, or spouse in their previous life?
Great movie! I opened this thread to mention it.
That’s exactly what Audrey Rose was about.
Exactly what The Mummy with Brendon Fraiser is about. Except nobody wants the reincarnated to actually get together.
The movie Chances Are is about a man who realizes he’s the reincarnation of his girlfriend’s father.
If you don’t mind a book about stories that are supposed to be true, Return from Heavenis a collection of stories from people who believe their children are the reincarnation of deceased relatives.