I'd expect the first exploration to be done by machines, and digital transfer of the controlling instance would remove almost all drift.

Machines suffer mutation rate too. Cosmic ray induced bitflips could be possible. Although since we are all spitballing anyhow maybe you can handwave a cosmic shield along with your cosmic explorer.

It would also be interesting if the host system collapsed. That would be some interesting scifi fodder: advanced civilization sends out probes but by the time FTL visitors show up, their civilization already collapsed to the stone age.