This is an interesting question. There is a story in the scriptures about the Buddha and a wild elephant. I don't know whether any of you have heard this:
There was a wild elephant that was unable to be trained and Devadatta, the Buddha's cousin, was planning to use this wild elephant to trample the Buddha. Due to intense jealously, Devadatta didn't like the Buddha. So Devadatta let this wild elephant loose, forcing it towards the Buddha. Through the power of the Buddha's Compassion and Lovingkindness, before the elephant got to him, it went down on its knees in front of him and was tamed. This was the power of The Buddha's Lovingkindness that could be felt by others. Even if the Buddha's Compassion and Lovingkindness can not purify the elephant, the Buddha did have the power to influence other beings, and obviously this elephant. It is taught that we have to purify our own mind. However, the influence of someone with strong Compassion and Lovingkindness is great, so that another being can be inspired to develop that within themselves.
It is also believed in Thailand that if people get together and have strong good wishes for others, that it can help some other people. Whether it's because the other person knows that everybody is doing it for them and then it brings a confidence and a good response in them, I am not sure. One time, our former abbot was very ill, unconscious in hospital, and they thought he was going to die. Due to a liver problem, he had lost a lot of blood. All of us got together and had a chanting session directing Compassion/Lovingkindness towards him.
He was going downhill and then he recovered. I am not sure whether we had any influence on him or not, but we certainly were wishing him well. I don't believe that he would have recovered unless he had the Kamma to recover. However, it does allow the people who are wishing it to feel that they are doing something useful for the other person. We did transferring of merits. This is when we do some good Kamma and we wish for the good Kamma to be transferred to another person.
I am not too sure whether it actually helped, due to not knowing the power of these things on that level. But certainly we can't purify another person because everyone has to purify themselves. However, I do believe good wishes can help them.