Are you talking about academic misconduct (cheating) or simply poor performance?
With respec to to cheating, policies vary from institution to institution. However, in my experience it is common in the US to have a central authority responsible for recording and reviewing cases of academic dishonesty.
For example, on my campus faculty are required to report incidents of academic dishonesty (cheating, plagiarism, forged signatures on registration documents, etc.) to the Associate Vice President for Academic affairs. Individual faculty can punish students by assignment of failing grades, up to and including failing the course. There is a formal process by which students can appeal these decisions. However, if a student has repeatedly violated the policy they can be punished further by suspension or expulsion from the institution.
A huge advantage of this approach is that if this were left up to individual instructors a student could cheat repeatedly in multiple classes and never suffer any punishment worse than a failing grade in an individual class. In the other direction, it provides students with due process if they think that a faculty member has acted improperly.
A suspension or expulsion for disciplinary reasons will typically remain on the student's transcript forever.
If you're talking about poor academic performance (low grades), that can be a very different issue.
It is certainly the case that poor academic performance in the past (having "flunked out" of a program) can affect your chances of being admitted to another program even many years later. The application process nearly always requires the student to provide transcripts from all colleges that they have attended.
In the US, a student who has "flunked out" typically can apply for readmission to the university after some period of time. Many institutions have policies that explicitly disregard academic failures (and the associated low GPA) that happened in the distant past e.g. a student who flunked out 5 years ago or more may be readmitted and start with a new GPA. However, under such policies the student's transcript will still typically show the earlier academic failure.