Currently, when a remote capship is mining or gathering cargo, it will just stop dead in its tracks when cargo is full.

A nice quality of life improvement to this would be to add a notification like the "shields up" notification for when the cargo bay is full. I.E, a remote goliath is mining, it fills its cargo bay.
"* Forward Unto Dawn's cargo bay is full!"

Edit: This could be taken further, regarding mining with a notification when the capship has stopped mining. This is likely far more difficult than it sounds, though. I've been having some weird mining stops today, before cargo is full. It would be nice to know when this happens.

We'll take a look at this, and see what we can do.

This has been rolled a few hours ago in tonight's release. This release was entirely server-side and has no patch (or patch-notes). It'll be included in the patch notes for the next release.

To put it simply, remotely-commanded player capships will now report when they stop mining, which may be for a variety of reasons, including cargo capacity.