Lets say you have a Fireteam core team with a member who has Haris. If the Fireteam drops to 3 members including the Haris skill can I create a new Fireteam core and keep the existing Fireteam? Follow up question if the answer is yes: If I have a Fireteam Core with 4 or members and a Haris member, can I create a new fireteam core, and downgrade the old core to a Haris, dropping extra members. In the Druze sectoral this has some potential.
You have to spend a command token to reform the old core fireteam as a haris fireteam. So yes to both, but you have to pay for it each time.
Yea, it has to do with creating and canceling a Fireteam. To add some more color to Commoner's answer: You can't ever "downgrade" a Fireteam. You can cancel Fireteams and you can create Fireteams. In deployment, you can create Fireteams for free. After that, you have to spend a Command token to do it. You can cancel a Fireteam voluntarily whenever you like, for free. You can create whatever Fireteam you want so long as you meet all of the requirements for it (but again, after Deployment it costs you 1 Command token to create a Fireteam). So in your examples, if one member of a 4 member Fireteam: Core leaves the team, you probably still have a Fireteam: Core unless a cancelation clause was met, such as the Team Leader entering a Null or Isolated state. You can voluntarily cancel the Fireteam: Core, and then spend a Command token to create a Fireteam: Haris, because your three models meet the requirements for creating a Fireteam: Haris. If you already had a Fireteam: Haris on the table, creating the new one would automatically cancel the existing one. In the second example, you can cancel the Fireteam: Core for free, and then spend a Command token to create a Fireteam: Haris with the Haris skill model and 1-2 other valid models (assuming they are all in coherency of the newly minted Team Leader). You can then spend a second Command token to create a new Fireteam: Core. Alternatively, you could spend a Command token to create the new Fireteam: Core, which would automatically cancel the existing Fireteam: Core, and then spend a second Command token to create a Fireteam: Haris out of the models that had been in the original Fireteam: Core. Six in one, half a dozen in the other. The salient point is that Fireteams must be canceled and created. If you meet a cancelation clause, no more Fireteam. If you meet all of the requirements to create a Fireteam, you can spend a Command token to do so.
Also keep in mind that a team member who got kicked out of the team for whatever reason must re-enter it during the next order count. For example, a trooper who declares a minority ARO but remains fully operational and within ZoC of team leader. It's not directly related to the topic, but tangentially
http://infinitythewiki.com/en/Fireteam_Integrity "f the Fireteam has not been cancelled, then the troopers who left will automatically be able to rejoin it if they are again in Coherency with the Team Leader during the Order Count phase of their next Active Turn." Because of this line in the rules, the notion that a Fireteam: Core can drop to Fireteam: Haris if enough members leave it becomes moot - it can not happen. Only in the circumstance where a trooper remains outside of a Fireteam (such as keeps on being ISO, UNC or plain old Dead) would a Fireteam actually be permanently reduced in membership potential. (Clearing states obviously necessitates those states being contracted during Active turn since Doctor and Engineer can not be used in ARO) But what I'm pointing out is that just because a model is out of a Fireteam doesn't mean it won't get back in again, so a Core must by necessity remain a Core until the Leader is kicked out or the only one left. No2 L2 as was suggested won't work unless it's got significant complications attached.
It must or it might? Because for me "will be able to rejoin it" means that I, as a controlling player can actually chose whether or not he rejoins it.
It doesn't matter, because if you will run into a problem unless you plug this hole or you will run into cases where people say they have permission using the integrity rules to take a Haris up to 4+ members.