Maybe it is worth just putting out there (i.e. expressing) what we ourselves initially think about compassion?
The word itself (“compassion”) for me has a more global quality than the word “love”. Of course, these are just words, and one can give them any meaning one wants to - and K himself often used these words interchangeably - but love, the word “love”, seems to me to be more personal; while I take it that “compassion” refers to the ‘whole situation’ as it were.
Although K sometimes said that compassion is not a feeling, in my understanding at least compassion does involve a feeling aspect (“passion” means suffering after all).
According to my etymological dictionary “passion” in Middle English meant "the state of being affected or acted upon by something external”. Of course we know that K often took a common word and gave it a new meaning, but the word itself usually connotes a feeling aspect.
K often paired together compassion and intelligence as two fundamental qualities that a person must have to perceive truth. Putting it crudely, intelligence has something to do with the acme of the mind, while compassion is perhaps the acme of the heart (even though these two qualities are not fundamentally separable in K’s teaching - for him the acme of intelligence is compassion).
So compassion - in my mind - is something like a feeling or heart-quality with respect to the whole: to what is happening in human society, the ‘feeling response’ we might have to poverty and degradation, to the suffering of ignorance in every clime, the tragedy of heart-break and loneliness all around us; to the thoughtless killing of other human beings in war, the pain and grief this creates for thousands and thousands of people all over the world.
And also for nature - our response to the fact of how we human beings are despoiling the natural world, cutting down forests, killing animals on an industrial scale, for profit, for sport, for pleasure.
So compassion - in my understanding - involves our response to (and insight into) to the global suffering of others (of which we are a part).
However, compassion is also - as I understand K’s usage - a quality or state that is, in itself, entirely free from suffering; that can expand the horizon of our perception to include the whole world of beauty, perhaps even the whole universe in its scope! So it has a mystery to it too. It is perhaps an energy by itself, without reference to how I myself (or you or another) feels about the world of human suffering.
In K’s language compassion is a passion - an energetic feeling-perception - for everything: for clouds, for the blue sky, for the stars at night, for trees and people and “everything the earth contains.”
That intelligence which is supreme is everywhere. It is that intelligence that moves the earth and the heavens and the stars, because that is compassion. (Mind Without Measure)