Past lives research do not go against religion. There are some religions out there that believe that there was a past life and people today have been reincarnated from a previous life. So that means that research in a past life directly ties into a religion in on way or another.
Most religions have some kind of God involved and God is usually seen as all powerful so who is the average human to say how many lives God gave us and he definitely wants us to have as much knowledge as possible so there should be no conflict at all between religion and past life reseaerch.
Researching past lives can most definitely go against the tenants of some religions, but obviously not all of them. Reincarnation seems to have originated from Hinduism, where they have many books and beliefs dedicated to the subject. Where as, Christianity seems to center around a one life ideal. Therefore, it really depends on what religion you believe in, rather or not studying past lives goes against your religion. It certainly doesn't go against all of them.
Supporting a religion is all about having enough faith to look beyond what science tells you and believe in the unbelievable and often unsupported teachings of organized religion. Enough faith in the religious teaching which have been offered will allow you to look at the research into past lives being nothing more then an interesting article on your Internet browser. Assuming, of course, you are permitted by your religion to use modern technology and view material which is not approved by your religious leaders.
Why does looking at the past lives of family members or others across the world go against the teaching of religion? I don't believe that religion has anything against people looking at what has happened in their family line. Things like this are doing nothing but, teaching people nothing less.