Hello Jonatan,
Yes this is possible. You can write a plugin to set process id of at the creation of a record. Of course plugin can be triggered on update of fields to meet your conditions/rules also.
If you are working with Dynamics 365 latest release, there is an entity created for every BPF. Also, a record is created for every record on which BPF is available. You can write some custom code to update Process id and Stage id in that record.
This is sample to achieve it.
https://community.dynamics.com/crm/b/develop1/archive/2013/09/20/how-to-change-process-and-stage-programmatically
Jugal Kishore Dandamudi.