Avatar support when moving hosts / changing domains

[jeffpaul]

Is your enhancement related to a problem? Please describe.
Originally reported on WordPress.org support forum:

After moving host / changing domain the avatars will still use the OLD URLs. I suppose a nice way to circumvent this were to store the attachment ID instead and get the URLs of the resized versions from that.

Describe the solution you'd like
TBD

Designs
n/a

Describe alternatives you've considered
none

Additional context
n/a

[faisal-alvi]

Reopening after the #216's revert.

Require a robust solution to this issue that does not have regressions, especially like #245 and #246.