Anytime you change job classifications (e.g. PT->FT, PT->PT air, PT->OMS, etc.) UPS performs a background check. Additionally, if you're applying for a managerial-level position, UPS will perform a credit check.
We have a FT driver sup who spent his 20s behind bars for manslaughter, began his UPS career as part of a work release program, got his degree upon getting out & was quickly promoted soon after. He runs church groups, speaks about his past, etc. and thus has demonstrated that he's changed. But there's people who would've held his past against him. It all depends on your current management team. Of course, if your felonies legally prohibit you from doing the job (e.g. you're a sex offender who's prohibited from being around children unsupervised).