Assuming the capsules followed planned trajectories, they could have used technology such as that which is being developed in the Breakthrough Starshot initiative, described here: https://breakthroughinitiatives.org/initiative/3
It's so much easier to build and train AI then migrate our "humanity" into machines and just go from there to the Tannhäuser Gate and beyond.
When I was a kid, my dad, a very spiritual guy, after tucking me in, revealed to me: "Son, humans can already travel faster than light". My young and inquisitive scientific self said "What?? How?!?!" And he goes: "Just close your eyes and imagine yourself on Mars. It's instantaneous."
Plus - why only one source of these capsules? Does only one mold cell in a colony generate a spore? Drake suggests that perhaps many will have done this same thing.
The capsules will go around the galaxy, travelling through or oscillating above and below the galactic plane just like other stars.
Bonus 1: Make your capsules charged so they're guided by galactic magnetic field lines into star-forming nebula where a high density of stars are created letting them mix into protoplanetary disks.
Bonus 2: Just keep spewing out more capsules to increase mixing rates.