Mindfulness and compassion are often talked of as "two wings of a bird" by dharma teachers - mindfulness doesn't automatically translate to compassion. This sutta makes the same point: Maṇibhadda Sutta (SN 10:4) On one occasion the Blessed One was staying among the Magadhans at the Jewel-stand Shrine, the haunt of the yakkha-spirit, Maṇibhadda [Auspicious Jewel]. Then Maṇibhadda the yakkha-spirit went to the Blessed…
My issue with this specific concept is that lots and lots of people are lazy and instead of doing compassionate things, they "feel compassion" as a generic emotion through the use of meditation. I studied Buddhism quite a while pretty seriously, and I know that a lot of the meditations involve practicing loving yourself, and your friends, family, community, and that these feelings of love are a core thing to have in…
What did help me though was a book called “Constructive Living” by David K Reynolds. It talks a lot about what you mention. The importance of “doing what needs to be done”.
I got quite into meditating, but I found it just made me go far too inside my own head. It’s easy to feel compassion inside your own house.