I never experienced any magic in the timing, just don't try two consecutive workouts without a meal inbetween. Running is not necessarily very glycolytic unless it's high speed. Not sure about karate.
I find 50~150g carb is a good range for most moderate forms of exercise. If I'm dragging ass and can't improve then I'll nudge up the carbs--preferably with an equal mass of starchy and sweet plants, and maybe half-portions of cooking and dressing fat to compensate. Keep the calorie deficit small though--if you reach the point of light-headedness or insomnia it's not really helping.
Lots of: urban hiking, cycling, sprinting
Lots of: fresh meat, seafood, eggs, organs, tubers, starchy fruits, low-energy plants, meat fat, dairy fat, oily fruits
Some: cured meat, dairy protein, sweet fruits, rice, peas, tree nuts, oily seeds
Minimal: soy, refined proteins, sugar, liquid carbohydrate, grains, beans, refined oils, peanuts