Many of the checks that were available in ArcMap can only be used in Pro as constraints when creating or editing a feature class. For example, feature integrity checks. The only way to run them using Pro on existing data is to author a batch job in ArcMap and run it in Pro.
Currently, there are no plans to enable creation or editing of Data Reviewer Batch Jobs in ArcGIS Pro.
That said, there are authoring capabilities available for some (unfortunately not all) check types using map-based rules. Here are links to a series of quick-start tutorials that outline authoring and validation workflows using map-based rules.
Long-term, the Data Reviewer team is working to bring additional checks to Attribute Rule-based workflows. For the ArcGIS Pro 2.5 and Enterprise 10.8 releases, a number of Data Reviewer checks will now support validation workflows that leverage Server's validation server capability.
Soon, we'll also be publishing a Road Map (similar to what is publish for ArcGIS Pro) on GeoNet that will outline migration of automated validation capabilities to ArcGIS Pro and Enterprise.
Hmmm, I'm just starting to investigate validation in Pro and so far my conclusion is that I cannot have a Data Reviewer Batch Job for branch versioning (ArcMap cannot utilize the branch versioning-enabled connection file so I can't write one).
I *want* to like validation rules stored in the geodatabase, but I think my validation rules would be pretty extensive as legacy data from different time periods will likely need to be handled differently - our current data reviewer batch jobs and map-based reviewer rules allow for a severity (error vs. warning in our ArcMap environment). And since the rules can only be saved in a Pro project/map, I would need to share that project/map for multiple editors to use the same set of rules? I've probably missed something... so, looking forward to seeing the "road map" that Jay Cary mentioned.