As Amber said, every center does things differently.
At Stanford, you followup with the tx team post tx and don't go back to the CF clinic, at least for adults. Everything from tx forward has to be cross checked with your tx team. Every single medication you take, OTC & prescription. You will and can be followed by other specialists for other non tx issues but most things will need to be "approved" by your team. For example, you'll go to a dermatologist yearly, well woman exams if your female, endocrine (if needed) etc. But, say you have chronic pain issues and go to pain management (not necessarily tx related, say migraines) and they want to do a treatment, depending on that treatment (say injections of some sort), it would have to be "approved" by the tx team.
We were surprised when they said, even eye drops had to be approved. It all sounds intimidating but only at first. Once you know what you can take/use, it's not so bad. It's a learning process. Now, if your normal location is not where your tx center is, then yes, you are followed by your CF team who works closely with your tx team. This was how it was done for my daughter. We/she, did not live near her tx center and so was seen by her CF docs, who then reported and worked closely with her tx team, on every single issue. We/she were very adamant about following the rules on that, after all, she wanted to take the best care of her gift that she could.
BTW, if your CF clinic doesn't require wearing a mask, my post tx advice: ALWAYS wear at least a P95 when you enter a hospital or clinic environment. That yellow paper mask was referred to as a "show" mask by her tx clinic. My other daughter uses a P95 when she goes to clinic or the hospital even though she's not a tx patient.
As for post tx life. Free! Life is free! Take advantage of your new lungs and do everything you can! I've been told by many post transplantees, that they hated running but did so now because they can! What you can and can't do depends on your team. Tatoo's post tx is a giant NO! by just about every team I've heard. Stanford for sure! Scuba diving, snorkling... go for it! Although my daughter team didn't like the idea of sky diving, they didn't forbid it. Public hot tubs are a no no but a privately owned one where you know it's clean and properly maintained was ok.
There's a lot of oral meds to take, but it's all manageable. My daughter said, she'd rather take 30 pills a day in trade for breathing treatments and PT any day. There's a lot to learn when it comes to meds, but you will learn all of the names, brand and generic. It's amazing what you will know by the time you get through the process. As a mom, I started to feel like we should have an honorary degree with all the knowledge we absorb about meds and side effects and interactions (PILLS). I'm sure I didn't fully answer your questions but I hope that I've helped. I hope your kids are doing well and that tx is not in your near future and you're just wondering.