On your screenshot, I can see that all your Champions are already busy with other quests, it's indicated by the signs of compas on their pictures. The other available Champions don't meet all the necessary requirements to participate. 

The Champions will be found in the other quest that you have  done if you check through them  you will see where they are being used . As  you can see even low  level Quest can use high level champions at times.  

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Click the arrow on the right of completed quest to see where your champiom might be.

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 If it's completed  Claim it and it will free up your  champion for another Quest . Edited to include Please note  this champion says Minumum rank  this means that higher ranks can also be used if there are  no 4 or 5 star it will use 6 0r even as high as 8 is you have one.

I don't think you need to claim them to free up the champion, they should become available automatically once the quest is completed.  I have a bunch of completed rank 6 and 7 quests stored up and the champs seem to still be available.  

I have personally done this many times you can only use a champion once until you claim the quest it was used in it as this player doesn''t seem to have as large of a pool to use it would be easy to run out if quests remain unclaimed

Hello lord krasybg , 

Lord Wesley has right ,

you should to claim some quests for have free  champions , and after that you can use them in other new quests ... Lord Wesley explained very well how to check where the champions are ...

I wish you good luck and a good day !

Oh you can make a test check for see you are wrong ... 

Just when you claim the quest than the champion is free to be used in another new quest 😎