I applied through a campus interview process. The campus interview was pretty much behavioral based questions. Nothing unexpected. Since I was applying for a PhD level position, so they expected you to relate your answers to you research experience to some extend. This stage was just making sure that you were not a book-smart nerd.
If you pass this round, your info would be forwarded into exxon internal system. If a relevant position opens up, you will be contacted. I was contacted after about a month for a phone interview. I spoke to two research engineers. One of them was a senior engineer and the other one was team lead. Again, a lot of behavioral based questions. Make sure you are confident enough when answering questions. A lot of these questions can be found online, make sure you prepare for it.
A week after, a onsite interview was scheduled at Exxon houston campus. Flew in a night before, had a dinner with a division manager and a senior engineer who was the host of my onsite interview. On the next day, I did a presentation on my PhD research and there followed 6 1-1 45mins interviews with research engineers, team lead, and a division manager. The interviews consisted of both behavioral questions and technical questions. Again, nothing unexpected, but they do look for high-standard answers. Like challenges in your research, leadership experiences, and innovation of your work. I guess just be yourself and be confident. It was a long/full interview day. I got an offer after a month. I knew they were interviewing another candidate.
Overall, it was a very good experience. You got to see the new exxon campus and learn their culture. I guess the only tip I can give is to think about what exxon expects the potential employees to be like: thinking out of box, strong problem-solving skills, leadership potentials, stuff like that.