I think liberals and conservatives honestly define the word differently. For a liberal, compassion is about guilt. You should 'check your privilege' (ie, feel guilty), and act out of acknowledging that. And yet, that makes compassion about you, not the person/people you're supposed to be helping. Your attention is, by the nature of guilt, split between making yourself feel good and actually doing good. (General you, here.) Only by doing good or appealing so some social pressure can you alleviate your guilt, which compromises your actions to help others. Therefore, guilt is useless in helping others.
Yes, I am saying we shouldn't feel bad/guilted for the people we help. It's not necessary. It's not helpful. And frankly, it can be condescending.
In the conservative definition, compassion is defined as action. It is doing the most good in the most effective way possible. It sounds cold. It's not about your feelings. And we certainly do live in the age of feelings! But you can actually do people a wonderful, amazing amount of good by looking at a situation objectively, rather than with guilt, which clouds your judgment into making it about you, and how you feel.
Liberals think, "What feels good? How bad does this make me feel, and what does that say about me?" and conservatives think, "What will work?"
So what is compassion then? I'd say it's love. Love isn't just a feeling, an emotional reaction. Love is choices made and actions done. Think about your relationship with your family and peers. It's love that makes you kind to them, not guilt. Why should anyone else have less?
TL;DR: Compassion is love, not guilt.
[edited to clarify]