They may ask you to bring a resume or describe your previous experience and why you are interested in their work. I am changing colleges but want to still work with the mentor from my old college. Yes, working across disciplines is not a problem at all. If you are thinking about doing an Honor's thesis, you might want to check with your new major about their requirements regarding whether your research mentor needs to be in your department or college. In most cases, students will work during the summer on their academic year project.
Summer can also be a time to take advantage of unique opportunities.
For example, you may want to work with a different mentor or you may find a research position off campus. The ESR makes it possible for you to take advantage of a summer research experience in which you won't make enough money to cover the student contribution that Cornell expects of all students. You would need to plan ahead for which summer you want to use the ESR and it is only eligible to students who work over 280 hours during the summer. Ask your faculty mentor for suggestions. They may have a colleague who is working on related research or a slightly different aspect of your project.