yes they do!
More then just Beta particles off the sun, thermonuclear processes so energetic heavy particles!
The atmosphere shields but not completely, the earth’s magnetic field re-directs charged particles (beta, proton, etc.), of course how much defection depends on their energy. Also there earth’s field is not static but dynamic, strengthens and weakness even goes to zero & flips polarity. (There is evidence that is now happening!) Also the field is not simple, earth’s field is not like a simple bar magnet there are other components project another field in a different direction much weaker but there.
As you said fluid dynamics governs the core, but not a true fluid has plastic qualities, also has internal currents, dynamics magnetic field therefore its electromagnetic fluid dynamics or MHD - magnetohydrodynamics. All very very non-linear mathematics very hard to understand it well.
Yet it all works for us and makes life possible on the earth.
Very true, the field changes but very infrequently relative to human existence. And when it changes, dramatic things happen on Earth.
Sorry if I am repeating myself here...
I am suggesting here that, as with Sun photon radiation which affects (heats the Earth and interacts with the atmosphere), so does the Sun’s mag field interact with Earths. And the Sun mag field is many times that of Earths. I am even suggesting that the Suns mag field induces the Earth’s mag field (maybe cancels it out if it is self induced).
Hey, there is a thought — does the Sun’s mag field affect the Earth’s orbit (in additional to gravity) and might the earth’s orbit radius shrink or expand based on this attraction. However, if the Earth’s mag field is a result of the Sun’s mag field, then equilibrium is reached. And, if the Earth’s mag field is independent of the Sun then the two bodies can attract or repel. I am guessing that the former is true. :)