I could be wrong, but it sounds like they want to do the background search because they believe that, given the ex's previous choices, it would be beneficial because the ex makes poor choices in partners.
I can understand that viewpoint completely, however, I don't think that investigating the new bf is appropriate unless they have a tangible reason to believe that the ex's new bf could be harmful to the kids. The ex screwing up the last time in choosing a bf does not mean that this bf will be bad.
[em]Lucky
Lead your life so you wouldn't be ashamed to sell the family parrot to the town gossip.
- Will Rogers[em]