This is a difficult topic for me also. Here’s the personal info I’ve gathered so far…
Running short distances with a zone 1 heart rate doesn’t seem to affect my climbing the next day (assuming cragging). However, running hard or long the day before climbing definitely hurts my climbing performance the next day.
My question would be this: How much running and what heart rate zones (aka volume and intensity) are good to target if I’m cragging midweek and climbing alpine rock on the weekends?