What is the best way to use CiviReports to show how many members changed from grace -> expired in a given month (summary), not detail 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&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
Membership Summary report doesn't provide the information you're looking for out of the box. Is there a reason that using the Membership Detail report or Activity report and just using the "Statistics" section at the bottom doesn't meet your needs?
If you're committed to this path, you can write a custom report based on the existing Membership Detail or Activity report that doesn't show any results, just the "Statistics" section.