Hello @mrym - Welcome & thanks for posting on MS Q&A! Could you clarify the order of steps described in your statement below?
If you 'publish' from an Azure project or 'swap' from a 'deployment slot', the following error will occur if you make a request to an already deployed function before the function app is fully deployed.
Blue/green deployment pattern for Functions is indeed achieved through the use of deployment slots, and once configured correctly & between swaps, there shouldn't be any outage though.
Referencing the following resources as they may be helpful: