Gmail enforces strict daily bandwidth limits to protect its infrastructure and prevent abuse. This restriction can happen during intensive syncing, large mailbox downloads, or frequent send/receive actions. If your account exceeds the number of requests allowed for that period of time, Gmail will apply a limitation known as “throttling”. The restriction is only temporary and will be lifted automatically after a day.

You can also try to restrict the amount of data downloaded into eM Client.
Go to Accounts and under the IMAP tab of the affected Google account, disable download for offline use or limit the time period of messages you synchronize to decrease the amount of data downloaded at the same time, at least temporarily. 
You can try increasing the time period over the following days if you need the full message history downloaded locally.

If throttling continues for an extended period of time even with these precautions, there might be additional issues on your email provider's side. In that case, please report this to the Google support team so they can look into it or reconsider these limitations. 

How can you tell if your account is being throttled?

Throttling is usually visible in the IMAP diagnostic logs. A snippet of what it looks like in the logs:

01: A3 OK Success [THROTTLED]

01: A4 NAMESPACE

01: * NAMESPACE (("" "/")) NIL NIL

01: A4 OK Success [THROTTLED]

01: Detected server type: Gmail

01: A5 ENABLE CONDSTORE

01: * ENABLED CONDSTORE

01: A5 OK Success [THROTTLED]