I use both Thunderbird and K9-mail as email client, and while Thunderbird fetches email (i suspect it uses a lower threshold than K9-mail) i have to wait for getting email on my smartphone with k9-mail, as it uses a timer to fetch emails. Does Runbox supports IMAP IDLE?
Yes, our IMAP servers have IMAP IDLE and the last time I checked it does work as expected. Have you tried using it?
Unrelated to the main issues, but K9-mail is no longer supported by the developer. I recommend switching to a more actively updated program.
I checked and on Thunderbird the check for message when available is enabled, also it searches every 10 minutes.
K9-mail seems to not support IMAP IDLE and seems to support it from version 5.800, also it uses a 15 minute timeout for checking email, the lower timeout possible.
I tried Fair Mail and i don’t liked it. I’ll wait for a new version of K9-mail or wait for email or manual check for them.