Moving adult family sharing member to their own account.

It should work automatically, yes. "Should" being the operative word.

The Family Member subscribes to their own iCloud+ storage while still part of Family Sharing.

Once the personal subscription is active, you can remove them from Family Sharing and all their files, photos, backups etc.. will continue to be available to them now on their own iCloud storage plan.

There's nothing that needs be done beyond assuring the Family Member has their own storage plan before leaving Family Sharing.

There is no transfer of files or data, simply because there is no other iCloud location to transfer to or from. iCloud is a single location on an Apple server. You're just sharing storage space. So when he leaves family sharing, the system just continues to access his files and data automatically and the system will release the family member's used storage on your plan automatically. The data and files are not on your account or inside your file area at all. The space they take up is just dynamically counted against your subscription plan's available storage space. When the family member leaves family sharing, the space their files and data take up is now counted against their own subscription. The files and data have not changed locations at all. They still belong to the Family member and are under their account.

Limnos is just suggesting they can leave the Family Sharing group before getting his own storage plan, and while that may be possible, its better not to chance losing anything, by getting the individual storage plan before leaving Family Sharing. That's it.

If they have not taken out their own subscription by the time they leave the family their iCloud storage will be frozen for storing additional files or doing backups. If they have started their own subscription (even if covered by yours for a bit) then it should be as if nothing had changed, except they will now be paying for their own subscription. Apple's probably doesn't actually physically move the files, they just flag them as being part of a different subscription.

Again, even if the person does not start their own subscription before leaving the family, I have never in 20 years seen anybody post saying they lost their files because they had not been covered by a subscription for a short while. Yes, eventually Apple would likely do more than just freeze them.

This is turning into a long topic over something very, very simple, particularly since the person seems to need at least 150GB storage. Have the person take out their own subscription before leaving. Have them leave the family and that will be it.

The individual should take out their own subscription to the services required, then leave the family sharing group. It is particularly important that they do that before leaving the group if they are benefiting from an Apple Music subscription. If they leave the family before they have their own subscription they will lose access to Apple Music services which means all their playlists, etc.

You should also read: How to leave or remove a member from a Family Sharing group - Apple Support

Before leaving Family Sharing, the Family member would need to, on one of their devices, go onto Settings ➜ Account ➜ iCloud ➜ Storage ➜ Buy Storage to buy their own storage.

The first charge will be charged to the Family Organizer though.

Once they have subscribed to their own storage they will have access to both storages at the same time. No data transfer is necessary. Once they leave Family Sharing their data is then accessible on their own storage subscription.

This is all an automatic process. The only thing is to not lave Family Sharing before having the family member buy their own iCloud Storage.

About the iCloud storage, this is what Apple says in this support article Downgrade or cancel your iCloud+ plan - Apple Support

What happens when you decrease your storage plan

● If your iCloud storage exceeds your new storage plan capacity, iCloud won't sync or update your information, and iCloud backups won't complete. iCloud starts syncing again and iCloud backups resume when you increase your storage capacity or delete information to make more space.

To qualify a bit what Phil0124 says, unlike Apple Music, Apple does not say it will immediately delete files if you leave a family group and are therefore left using more space than that to which you get for free from Apple (5GB). Apple will simply stop the person from adding or changing them (and if the person needs less than 5 GB then they don't even have to worry about that because everybody with an Apple Account gets that much for free). I presume, however, that the individual will soon want to take out their own subscription (if they need more than 5GB) to accommodate their iCloud needs. They can either do that before they leave the family, or soon afterward. Apple is not clear as to how long a grace period you have, though I have seen some mention 30 days. Read this support article for instructions on how to upgrade to iCloud+ --> Buy iCloud+ or upgrade your current storage plan - Apple Support

This is really sounding like nit-picking to me but the only other small consideration in the decision as to when and if the person needs an iCloud+ subscription is if they are using other iCloud+ services. Review those here --> What is iCloud+? - Apple Support

Any files in icloud belong to the user. They will need to subscribe to iCloud at a level to cover their file storage use. There is nothing else to do as the files are the user's.