The movement has no qualms about targeting Republicans for ideological deviation. It's the exception to Reagan's 11th Commandment. The movement pays close attention to votes in Congress. And there is some evidence that the movement has been successful in electing like-minded legislators, and in keeping them in line.
The modern progressive movement also has no qualms about criticizing Democrats for too much compromise. Obama has almost as many critics from the left as from the right, notably on Obamacare (too watered down) and foreign policy (unmanned drones).
But liberals/progressives/the left do not take ownership of the Democratic Party. They seem to recognize that the Democratic Party is the party that they should like, but they put the onus on the party to come to them. It's the Democrats' fault because they have no spine. The Tea Party never lamented that the Republicans didn't have a backbone. The Tea Party decided it was the party's backbone. The Tea Party didn't wait for the Republicans to come to them. They put a rope around the party and pulled.
If there is any truth to the characterization that polarization today is mostly a consequence of the Republicans moving to the right, this must be a big part of the explanation. There is a movement invested in moving the Republicans, while the equivalent movement on the left, which I think it just as vibrant, seems like it has given up on moving the Democrats.