In Process or Under Process?

Something is "in process," not "under process." However, you can use the word "underway." Explanation provided by a TextRanch English expert. Some examples from the web: My passport application is under process at the Regional Passport Office.

Do we say in process or on process?

If you are in the process of doing something, you have started to do it and are still doing it. The administration is in the process of drawing up a peace plan. Her novel is in the process of being turned into a television series.

Is it in process or in progress?

These terms mean the same thing, and there is no difference between them in terms of actual usage. In progress is the more popular version at this point in history, by a factor of several times.