This is only a part answer
University of California Berkeley applies a probation when the cumulative GPA goes below 2.0 and are within grounds for dismissal. But, they give a semester and advise the student to:
The College of Letters and Science encourages students who have been placed on term probation to seek advising at the Office of Undergraduate Advising for help with schedule planning and to discuss any issues that may have caused poor results in the first place. Although this web site may be helpful to you, it is primarily designed to assist probationary students whose cumulative grade point average has fallen below a 2.0.
An example from Australia, Brisbane's Griffith University, it seems that a similar constraint may be applied:
The minimum standard for academic performance is a Grade Point
Average (GPA) of at least 3.0 for undergraduate students and at least
3.5 for postgraduate students. Students who meet this GPA standard
are said to be in ‘good academic standing’. If a student’s GPA is below
this level their performance is ‘unsatisfactory’.
The first time a student’s performance is below the required
standard, the student is placed on probation for the next semester.
The second time a student’s performance is below the required
standard, the student will normally be excluded.
In the Princeton article "The Effects of Academic Probation on College Success: Lending Students a Hand or Kicking Them While They Are Down?" (Fletcher and Tokmouline, 2010), found that often there is an improvement, but then their
findings also suggest that this short term boost in performance fades out over time and students who are on academic probation following their first semesters of college do not have higher rates of persistence or graduation.
They also found that an overall model (indeed a definitive answer) is hard to come by due to inconsistency of heterogeneous effects, particularly
pre-determined student characteristics as well as high school of origin.