I am not certain on this answer, but I do believe they have to re-associate. They can't keep communicating on the existing channel, because the AP is no longer using it, so the client must re-associate with the new channel.
However the client may not have to do a full re-authentication, because that data may be cached like a roaming event. So the move to the new channel should be reasonably fast and invisible to the client (assuming the client firmware has good roaming support).
From the below logs, it looks like you are experiencing a lot of interference.
The MR16 is not really a good choice in this environment, because it has no independent scanning radio. It can only do opportunistic scanning when it is not busy serving clients.
Also, due the the interference, it seems probably you need more AP(s) to counter the interference.
I would consider getting a minimum of MR33's to operate in this area, and at least one more than you do now. You could also consider using the MR53, which has much greater capability to handle difficult RF environments. If it was my money, I would be going for the MR53's to have the best chance of stablising the environment.
Another option would be to try and identify the source of interference. You'll probably need to get someone in to do an RF site survey to try and figure this out. Unfortunately a lot of the time it is caused by things outside of your control (such as other near by business).
Another option you could consider playing with if you don't have money for more APs, is to play with the RX-SOP level to try and limit the impact of interference. I would only proceed with playing with RX-SOP in desperation. It is very easy to make things worse.
https://documentation.meraki.com/MR/Radio_Settings/Receive_Start_of_Packet_(RX-SOP)
https://documentation.meraki.com/MR/Radio_Settings/RF_Profiles#RX-SOP