CiviCRM Membership Status Change Summary Report
When a Membership changes Status an Activity of type "Change Membership Status" is created, and for your example, has the Subject of "Status changed from Grace to Expired". These factors can be used either in Find Activities, or Report civicrm/report/instance/3, or in Drupal to make a View, and if we use Aggregation, to get summary statistics as requested in the question. 

I decided to give the latter a Proof of Concept and have attached it. IF this is useful to you (or anyone) please consider 'giving back' (and forward) by chipping in to our project to get CiviCRM working fully with Drupal 8 ;-) http://www.fuzion.co.nz/civicrm/contribute/transact?reset=1&amp;id=4.

Hmm can't see how to attach file so have put in the Fuzion Treasure Chest at http://www.fuzion.co.nz/blogs/getting-summary-stats-membership-status-changes-view

And in order to fully answer the question, if you use the Views CiviCRM Dashlet module described here you should be able to set it up on your dashboard https://civicrm.org/blogs/brandonferrell/drupal-views-civicrm-dashlets