Along with all of the above, which was great posts btw, you build in responsiveness to anything by being super responsive yourself. So, you have to really pay attention to putting on enough pressure to get a response from the hrose and reward that response. And the response that you reward is NOT bracing, it is GIVING. So, you don't reward the horse while he's still leaning on the bit. You hang in there with contact/pressure until he gives, THEN you reward his give with an even bigger one on your part.
Too often people put pressure on the mouth, the horse resists, or he moves that way a little but he's still fighting against the rein, so the human releases. The hrose just learned to fight the rein, 'cause that's what gets the human to give off the pressure.