hey, johnram,
i'm both an emdr therapist and am currently doing emdr w/ my own t. yes, the brain can continue processing between sessions. if it's too much, i think it's important to talk to your t about it. you may need a break from it to discuss the processing that's going on. the pace may need to be slowed down. if it's possible to be able to talk to your t between sessions, that might help as well. many emdr therapists recommend that you write what comes up for you between sessions as well. i found that writing can help get it out of me and onto the paper, like it's a safe holding place for what i'm experiencing.
i often have one session of emdr processing, while the next session is basically me talking about what i've processed, and we can set up new targets for the next time. please remember that this therapy, like any other, needs to go at your own pace, and that sometimes the therapist doesn't always know exactly what it is. the more details you give to your t, the easier it is for them to know how to proceed with you.
i hope any of this was helpful. if you have any questions, feel free to pm me. love and hugs,