Well, maybe "constant speed" is a bad wording.
An example:
We have 1000 stars within the 30 light year radius neighborhood. Traveling at Voyager 1 speed, it takes half a million years to do a 30 light year trip. [1]
Given another half a million years for bootstrapping at the destination for ten more probes, we can send ten more probes, 1 million years from the launch of the first one. At two million years we get 100 probes.
How many probes do we get in a 11 million years? 10^11 probes. We have 10^11 stars in the galaxy.
Dinosaurs roamed the earth aleady 200 million years ago, and died in the mass extinction 65 million years ago. 20 to 5 million years ago was the Miocene, when life was pretty close to modern already. [2]
It's really the nature of exponential growth that makes things seem instantaneous compared to celestial phenomena or evolution. Colonizing the galaxy, even with abysmally slow speeds, is possible, because of the magic of exponentials.
(Certainly this has a lot of assumption, like that you can bootstrap, or that the probe survives the trip. Anyway, optimal probes would be a lot faster and bigger, and there would be more of them. This was just to tie in with something concrete that we know we can do, as Voyager 1 really is out there traveling at 15 km/s, as are Voyager 2, Pioneers and New Horizons probes.)
1: https://www.wolframalpha.com/input/?i=+30+light+years+%2F+vo...
2: https://en.wikipedia.org/wiki/Miocene