Prevent Government Shutdowns Act of 2020 This bill provides continuing appropriations to prevent a government shutdown if any of the appropriations bills for a fiscal year have not been enacted before the fiscal year begins and continuing appropriations are not in effect. The bill also limits official travel, congressional recesses or adjournments, and the consideration of legislation that is unrelated to appropriations after the beginning of a fiscal year if both chambers of Congress have not passed the appropriations bills for the year.