The only ways I have come across the term "process" is in two different meanings. One is the phrase "process comment" which means shifting from the content to what is happening in the room. It is especially useful when seeing more than one person, like a family or a couple, so you check in with where you are after they've interacted.

The other meaning is as in the phrase "unprocessed material" meaning usually unconscious stuff that is bubbling up and clearly there, usually interfering. That implies that you need to "process" the material. What that means exactly depends upon the approach you take. The books you list are all psychoanalytically oriented so I guess that would mean making it more conscious and finding the roots, seeing if there is a possibility of change or not, things like that.

"Fighting the process" might just mean "resistance" which is standard in older forms of psychodynamic therapy. Newer forms like relational don't put much emphasis on that; they made a radical break from traditional forms of psychoanalysis. Why not just ask what it means to the therapist who is using it?