When's the right time to put your baby in the seat unit?
TL;DR You can pass to a seat unit pretty much anytime - but you should never put your child in a sitting position if it didn't sit up by itself (without help) already. With bucket seats, the optimal age is 6 months and more, but even there, you can help yourself with a wedged insert, creating a lie-flat.
You have a baby around 4-6 months (maybe less, maybe more), and the baby starts wanting to see the world better. It is either unhappy in the carrycot, tries to pull itself up (to sit up) - even though, with no real result yet... And you're wondering if it's safe and healthy to move from the pram or car seat/travel system to a proper pushchair seat.
The first thing to know is - there are two types of seat units that will make the answer to this question a bit different in each case. Firstly, your stroller seat can be a lie-flat, with an individually adjustable backrest and leg rest, and then, it can be a bucket seat unit (more about all the pros and cons in this other article) that is shaped and without an insert suitable from at least 5-6 months of age, not sooner.
With a lie-flat seat, the answer is YES, pretty much in any case and even for newborns. Of course, you always need to consider the stroller type, comfort level, padding, and all that, but for that, you don't need to be an expert - you can see if your baby is comfortable enough in the seat. In any case, the yes, the baby can be in the seta unit pretty much from birth, is valid only if you put it in the seat in the lie-flat position.
No baby that didn't sit up without help should be put in a sitting position.
Using the seat in the fully-reclined - not the sitting - position is very important! The fact that the baby didn't manage to sit up yet (without you trying to "help" means its spine and muscles are not ready. You can actually endanger your little one by straining the unprepared muscles by putting the baby up to sit - it will be happy to see something new, but it can end up with scoliosis or, in the "better" case, 20, 30 years from now with severe back pains.
So - if your stroller system/pushchair is a lie-flat, there's no problem moving on to the seat immediately when you feel the baby needs a change/needs to see more/doesn't have enough space in the bassinet... in the laying position, of course (until it sits up, then you can incline the seat as well 😊).
The second type of seat unit, the bucket-style, L-shaped seat, is a bit different in the fact that it is shaped even when reclined. Even with not-yet-sitting children, it can be used, BUT too tiny a baby inside will not be large enough to bend the knees where it should and will be too lost in it. That is why the bucket seat is suitable, YES, even for babies not sitting independently, BUT FROM approx. 5-6 months of age (usually, 6 months is the norm, for a larger/stronger baby, it can be quite sooner).
Even smaller babies can fit there IF it is a very small-size seat OR if you use a bucket-seat-flattening insert like the one for Babystyle Egg or Quinny cocoon. These have a wedge that makes the seat flatter for even newborn babies - and yeah, you can use them even with different brands.
So when can you pass from a carrycot/car seat to a pushchair seat? With a few precautions, and not putting a non-sitter in the sitting position, even very soon is OK and not a health hazard for your child.
There's, of course, the case of a third type - I mean, not a type, more of a feature to consider - and that is when the seat is NOT reclinable fully. That, in the case of a not-already-sitting-by-itself baby, means NO, you cannot pass to a seat unit yet, if your baby doesn't sit up without help. You go to such a buggy when your baby is around 1 year of age (and later), usually.__
P.S.I also struggled with "pram to pushchair change" when I was a first-time mother. I am too curious, I guess :-D So I did my research, asked pediatricians, looked up what physiotherapists and other experts say, and to add to it, went over many many prams and pushchairs - and the answer I got to is this. Protect your child, even from its own curiosity, and let it grow properly without trying to make it faster. Most babies truly sit up by themselves about 8-12 months if not "helped" and sat up by their parents (which is not really helping their health, though). This is the normal time - everything sooner is either a rocket-boosted baby or a too-eager to help mum or dad.