How to become a king of a friendly dynastic kingdom

Here's how it works: If you are their liege, the gains from the war are part of your kingdom. It doesn't matter if they're your king, duke, earl, or even mayor, bishop, or count.

To claim a duchy from somebody else through war (when you don't have a casus belli that lets you claim it directly):

1. Find out who* has a claim on it by opening the duchy information screen and clicking on "Claimants" to see who has a claim on that kingdom.

2. Invite one of the male claimants over to your court. (If nobody will come over, or there are no male heirs, you'll have to find another way to get that duchy. Note that if one of your courtiers marries a female claimant that you can't legally give a title to, that female claimant is not your vassal and the newly-conquered duchy will not become part of your kingdom, even if you give your male courtier a title.)

3. Give that claimant a landed title. Any landed title will do, even a city, church, or castle.

   Note that if you land the claimant a city, gaining him a Duchy will found a vassal merchant republic instead of gaining you a new feudal vassal if you are playing version 1.9 of the game or later. Merchant republics are generally considered awesome to have as vassals due to their revenue and tax bonuses, but this might disrupt an existing merchant republic that is already earning you revenue.

4. Go to war with the ruler of your choice, using the "Claim on duchy X" as your casus belli.

5. Once you win the war, that claimant will be your duke.

I'm guessing that this will also work with kingdoms if you're an emperor. So far, I've claimed two duchies this way: once by giving away a county to the claimant, and once by giving away a bishopric.

*As revealed in this other answer, you can't actually press the claim in war unless the claim is strong, or the claimant is a pretender (2nd or 3rd in line), or there's already another war going on over the title, or the current holder of the title is a woman (and you have a male claimant) or a child (so there's a regency). You can only use "any claimant with a pulse" if the title is disputed or the ruler is a child; you can only use "any male claimant with a pulse" if the ruler is female; otherwise, you'll need to pick up one of the pretenders or someone with a strong claim: you'll need to do slightly more research to find who they are, and they're often harder to invite to your court.

(Note: if there are three or fewer male claimants, then it's almost certain that they're the heir and pretenders. Just grab one and go on your way. Figuring out the exact claims is more important when there are four or more male claimants.)

You kill your own Duke. So long as the Duke's son (whichever) inherits his father's lands before his mother's there is no way you lose Jazira. The chance of the Duchess dying first is why you are getting the inheritance warnings. If she dies first there's a chance that her heir could end up inheriting Jazira as well when the Duke dies. With Orkney having Gavelkind succession, if she has enough holdings there's a chance both of her sons could be landed vassals of another realm when your Duke dies. If both sons are vassals of another country when the Duke dies, you'll lose Jazira for sure.

As Affine and Studoku point out, the reason your High Crown Authority isn't preventing this is because Jazira is not De Jure part of Byzantium. It only can prevent out-of-realm inheritance in the De Jure Empire. I don't understand why Primogeniture succession in Jazira would be giving the younger son Jazira while the older gets Orkney, that's just backwards. However, so long as the Duke dies before the Duchess, and before either son is landed, I can assure you the inheriting son (whoever that may be) will be your vassal.

I know you have said you don't care to have Orkney, but if you did, the solution would be to kill the Duke, then immediately begin plotting to kill the non-inheriting son. This would assure that Orkney went to the son that just became your Duke.

As you say, this is Crusader Kings. All things are possible through adequate murder.

If for some reason you wanted to achieve your goal without killing anyone (a strange concept, but bear with me): If you have a title to spare (even a barony will do) you can land the son that is the heir to Jazira. By giving him the barony, he becomes your vassal. When he inherits further titles those will also become a part of your realm, even if the titles he inherits from outside the realm are of higher rank than his first title. So long as he is landed in your realm, any lands he inherits become and/or remain a part of your realm.