Windows 10 Live mail and Kaspersky updates won't auto download

Aha! I'm not sure if this is the actual reason or not but support for Windows Essentials, which Live Mail is a part of, ended Jan 10th of this year.
I also noticed this on a help page to do with Live mail:

Some email service providers have moved to newer email protocols for improved security and reliability. These email service providers may no longer work with Windows Live Mail.

Click to expand...

Ref:
https://support.microsoft.com/en-gb/help/10740
Not exactly the same as your issue but it seems that Live Mail is on it's way out.

Have you considered using the Mail application that comes with Windows 10? I've used it myself and it's not half bad.

When either of the two apps fail to do the auto updating or mail collection is their any error...