I think MF had a baby seat and diaper bag for fake baby, and maybe convinced HB she didn't need anything else and there wasn't time to mess with her carrier and seat base. She was breastfeeding, so no need for bottles and formula. When mine were babies, I'd often just tuck a couple of diapers and a travel size wipes in my purse. I hated carrying diaper bags. (To tell the truth, I had a fishing vest, the kind with a bazillion pockets, and I would tuck away only what I absolutely needed, so I didn't even need a purse and had my hands completely free for the littles.)
I could see MF drugging her, as well, if they were just sitting in the car to visit and have a cup of coffee. I don't know how else she could have controlled or restrained her after she got in the car. I'm not convinced she used a weapon, with their long friendship I don't think HB would have believed she was really in danger unless MF actually hurt her or the baby. And I don't think that happened until they were away from the apartment complex.
All speculation... Hopefully, MF will confess and we'll learn exactly how it happened.