The first thing is to understand that when we are doing Compassion and Lovingkindness, we try not to feel, "I am giving this to you." "I am giving this, I am creating an energy inside me and giving all this to all these beings." We are trying to feel with, lessen the separation. In order to do this, it is very helpful to remember the Compassion side of just trying to reflect on the person's Dukkha.
Whenever you start to see the mind thinking of yourself when you are doing this, realizing that you have started to think of "how you relate to them." Try to let that go, and to just bring the mind back to the subject and the Dukkha of that being or person you are trying to contemplate. As this happens, you are contemplating Dukkha and you can often see the cause and then Compassion arises with the person, rather than you giving Compassion to this person.
Lovingkindness seems natural after this, in a much less egotistical way, "I have to give this to this person." There's just a feeling of wishing for the ending of this suffering. So it is important to feel with, rather than have the idea of Compassion and Lovingkindness, "I am going to be a beacon of Lovingkindness and give every being in the world all this feeling I have within myself," which creates a feeling of separation.
Another thing you can do is to ask yourself, "Have I still got more to do myself, am I free from greed, am I free from hatred, am I free from ignorance?" As we see that we have more to do, we realize that we also have Dukkha, and it is important not to elevate ourselves above others. Also you can reflect on Death, and ask whether you have gone beyond Death.
As well, it is helpful to understand that we just don't know the potential of another person. It's very, very difficult to judge from the outside. A Buddhist story which illustrates this is of a naked crazy woman at the time of the Buddha. Most people probably thought that she had no potential at all. In fact, she was mad, she was crazy, she wandered from village to village, just eating the scraps from people. People shooed her away because she was mad and crazy and naked.
One day the Buddha came to the village where she was, and she was drawn to the Buddha and wanted to get near him. There was a whole crowd of people around the Buddha, and the people on the outside wanted to shoo her away, "Go away, you're mad, you're naked." She didn't understand of course, she was drawn to the Buddha. And the Buddha saw her on the outskirts of the crowd, and said, "No, allow her to come near me." She came nearer, and in the presence of the Buddha she became aware of her madness, and aware of her nakedness, and she became ashamed. Then someone threw a cloth, and she covered up and the Buddha talked to her.
He talked to her about Dukkha, and the universality of Dukkha and the cycle of birth and Death. She became first level enlightened at that time. The story was that in the space of two days, she lost her husband, her two children, all of her extended family. And because of the grief of this she became crazy. The Buddha, because of his powers, saw this and was able to give her appropriate teaching to enlighten her mind. And she understood Dukkha, so she didn't have any resistance to it.