Quitting a PhD program and starting another one: who is the legal owner of my research?
Within limits, you most definitely can bring the old research with you and publish it. Unregulated digital data stored on a standard computer should be fine, but for example, there may be limitations on moving human subject data (even de-identified data) depending on what the IRB protocol said. If the research involved samples (e.g., chemical compounds) moving these may also be regulated.

You may or may not be able to use this previous work towards your dissertation. You will have to check with your new department.