What happens to extra-galactic rays when they arrive at the solar system? Quazars send baryons from other galaxies towards us which are deflected from by the local magnetosphere. The early solar system probably picked up many millions of extragalactic cosmic rays for every tonne of local atoms. Humans probably contain some baryons which are sourced from >3 billion light years away.
What happens to extragalactic cosmic rays when they arrive at the solar system/at a protoplanetary disk? Where do they go? Does a proton from that far away adopt a local electron and become hydrogen?
 A: 
What happens to extragalactic cosmic rays when they arrive at the solar system/at a protoplanetary disk?

Note that cosmic rays are generally considered to have energies in excess of ~109 eV.  In a 1(0.01) nT magnetic field, a 1 GeV proton would have a ~5.7(566) Gm gyroradius.  Note that the gradient scale length of the heliospheric boundaries are on the order of 100s of km to several Mm, i.e., much smaller than the typical cosmic ray gyroradius.  This is important because if the order were flipped, one might not expect many cosmic rays below a certain energy to penetrate far into the heliosphere.
These are the lowest energy cosmic rays, whereas at higher energy the gyroradius can exceed an astronomical unit, e.g., ~1015 eV proton in a 1 nT magnetic field has a gyroradius of >20,000 AU which is larger than the entire heliosphere (on the ram side with the interstellar medium at least).
So most cosmic rays above ~1 GeV that are ions that come from outside the heliosphere enter the heliosphere and begin to follow the standard single particle motions associated with the observed electric and magnetic fields of the system.

Does a proton from that far away adopt a local electron and become hydrogen?

No, this almost certainly does not happen.  A proton with at least 1 GeV of kinetic energy is relativistic and so the recombination cross-section is going to be tiny.  Much lower energy protons can recombine and form low energy neutral atoms that can enter the heliosphere unaffected now by the magnetic field.  We can detect these with spacecraft such as IBEX.

The early solar system probably picked up many millions of extragalactic cosmic rays for every tonne of local atoms.

I am not sure about that but one could presumably calculate such things by looking at the well documented and published cosmic ray spectrum (e.g., see figure below).  Note that the roll-off at low energies is actually due to solar transients like coronal mass ejection (CME) which lead to a relationship called the Forbush decrease that tends to follow the solar cycle.

