This is very important because often, yes, our Compassion/Lovingkindness will be blocked because we're not caring enough. We have a wish for a certain outcome to our wish. We want to change that person. So, definitely if you notice that you are wanting to change them, then it is correct to understand that they are the owner of their Kamma so that you can bring equanimity to support the Compassion/Lovingkindness.
Equanimity is a balancing agent for Compassion and Lovingkindness. In fact, for the Unselfish Emotions it's very difficult to separate these, just like you can't separate the Noble Eightfold Path because they are balancing agents. Without Equanimity, Compassion and Lovingkindness can fall into grief and aversion towards the person or the world because we don't accept that Dukkha exists and that we have limited power over our ability to change the world and others. So, yes, that's correct, the learning how to bring up Equanimity when we see that actually there's non-acceptance, that our Compassion has fallen into grief at that time.
Equanimity is also a balancing agent to Sympathetic Joy. So that we don't get too giddy and forget our true purpose or that Dukkha exists. Equanimity is a balancing agent to all the Brahma Viharas.
On the other hand, Compassion is a balance to Equanimity so that we don't fall into indifference. If we have too much Equanimity, we may not care enough to feel with this person, after all I can't change them so why do I even want to think about them? We need to be careful that our Equanimity doesn't fall into indifference here. We care with, we feel with, understanding they are a human being with ignorance. But we do not have the power to change the situation if we are not in charge of the causes of their Dukkha. So, acceptance that Dukkha does exist helps in strengthening our Compassion and Lovingkindness.