What happens is that the eccentricity of the orbit of the Earth around the Sun gets 'stronger' during a 'glaciation'. And in these periods the situations on Earth get much more unstable.
And just exactly why would our host star lessen its authority over our planet? - no, Earth's parabolic orbit is held on quite a firm leash. Neither is Solar output variation a factor - quite simply, our not-so-bright scientists and the not-so-prestigious Scientific Institutions that host them and churn them out, are rather uninformed when it comes to an Ice Age. As a matter of fact Earth Science is a grossly neglected subject. Astronomy, likewise, is so preoccupied with distant stuff that when it comes to Sun-Earth relations, myopia obscures the obvious...
sad but true.