This WTF Flow demonstrates how you can automatically update a stage of a BPF.
It requires referencing fields from the BPF entity record
The lookup field that associates the BPF to the record
Business Process Flow Instance ID
If you import the Flow package and try to run the Flow it will most likely not work because you won't have the custom entity of Application in your instance. Use this WTF Flow as an example for what can be achieved to update the stage of a BPF automatically.