There are scholarships for PhD students that work at any accredited university, sometimes in a particular zone. Example.
Applications may be submitted by full-time students who desire to study propose to pursue lighting education or research as part of their undergraduate, graduate or doctoral studies. Applications for the Alan Lucas Memorial Educational Scholarships may be made by those who will be a junior, senior, or graduate student in an accredited four year college or university located in Northern California (including San Luis Obispo, Fresno and north). All applications should be submitted before April 1. The scholarships to be awarded will be at least $1,500.00. The student must submit an application, a statement of purpose, a description of work in progress, transcripts, three recommendations and a scholar agreement form. Copies of all enclosed documents should be photo-copied for interested students.
Does something similar exist for postdocs or tenured professors? In particular, can an applicant for postdocs/tenured positions feasibly say "I will bring my own funding"? If the answer is "yes", is this a powerful advantage in the application process (I know it is for PhD students)?