When Processes are executing, or have recently finished executing, you can check on their status from within the process page or from the activity page
From within a Project, click Activity in the left hand navigation bar
On the Activity page you can see a process currently being executed.
You can see the history of the last 12 batches for each process by drilling down on the bar.
Reading the Process Batch Log
In batch activity you will see batch steps summarised according to the process type;
Merge New and Changed Data
Read, Transform, Delete and Insert
Read, Transform, Insert and Update
Merge All Data
Read, Transform, Delete, Insert and Update
Incrementally Load Data
Read, Transform, Insert
Read, Transform and Insert.
Double click to drill down to activity warnings or error messages
See an example of the activity page drill down here...
In addition to checking the Project Activity page, you may want an additional notification of process activity, check out how to enable alerts in Project Settings.