Feasible, but inefficient, and needlessly complicated. It costs more to change orbits to rendezvous with Chariclo, then change orbits again to rendezvous with Ersa, then to just go straight to Ersa. Why? Delta-V and inertia.
Your setup has a few problems, you say Chariclo 101 is going 60,000 mph, but relative to what? The Earth is going around the Sun at 66,600 mph, so you probably don't mean orbital velocity. From your drawing it looks like Ersa is making about 1/6th its orbit during the trip. Asteroid belt objects have an orbital period of about 4.5 years which means the trip will take about 275 days.
The details don't really matter, there's an asteroid whizzing by your starting point that will later whizz by your stopping point.
The problem with your plan is velocity is not the problem in space flight. Once you're up to speed in space you can coast along pretty much forever. Changing velocity is, and your plan just adds more changes. Not just absolute velocity, but also direction. This is known as delta-V. This means changing your orbit costs fuel, and each rendezvous requires delta-v. Even in the 22nd century delta-V matters (and if it doesn't, why do we need this convoluted plan?).
Then there's the issue with your spaceship. Any spaceship which can rendezvous with and land on Chariclo then ride it to Erza, can more easily rendezvous with and land on Erza. You need a spaceship large enough to carry all your people and all their supplies for the trip and mining colony. You need fuel to leave LEO and rendezvous with Charicio, plus fuel to leave Charicio and rendezvous with Erza. You need supplies for the trip. If you already have a spaceship large enough to carry everything and everyone to Charicio and wait until it reaches Erza, why not just take it all the way to Erza?
To sum up, the Chariclo plan means you have to...
- Leave LEO in a spaceship sufficiently large to carry all your people and consumables for the whole trip.
- Go to where Chariclo 101 will be.
- Change velocity and direction to match its orbit and rendezvous, which means you're going as fast as Chariclo.
- Find a suitable landing spot.
- Land on it.
- Set up camp.
- Wait and consume resources.
- Tear down camp.
- Lift off from it.
- Change velocity and direction to rendezvous with Ersa 772.
- Find a suitable landing spot.
- Land on Ersa 772.
- Set up camp again.
- Start mining.
All that extra changing in velocity to rendezvous and land on Chariclo is unnecessary. Just take a more efficient direct transfer orbit to Ersa in your spaceship which already has to hold everyone and everything and already needs the fuel to have the delta-V to reach an object which will pass Ersa.
- Leave LEO in a spaceship sufficiently large to carry all your people and consumables for the whole trip.
- Change velocity and direction to rendezvous with Ersa 772.
- Wait and consume resources.
- Find a suitable landing spot.
- Land on Ersa 772.
- Set up camp.
- Start mining.
This allows you to pick a more efficient, direct transfer orbit, and at the time of your choosing. It removes the cost and complexity of an extra rendezvous, landing, and liftoff.
This assumes there's no consumables on Chariclo. If there are (such as mineable air, water, fuel, and reaction mass) it might make the landing worthwhile. If you can "live off the land" for the 9 month trip it will greatly reduce the amount of supplies your spaceship must carry, and thus reduce its size and complexity.