As a general rule, the First Amendment protects public or government employees from adverse employment action based on their political activity or support. This is not an absolute rule, and there are exceptions, which most often depend on the employee's job and the activity. See Political Activity & Government Employees.
These are usually referred to as political discrimination cases.
Lexington, Kentucky employment lawyer Robert Abell represents individuals and employees in political discrimination cases; contact him at 859-254-7076.