Karate and Cross Country: Please combine them!

My dojo is in a YMCA (note: I mean the dojo I attend, not as in it belongs to me, I'm not the instructor).  There are classes for kids and a number of high school age students who rotate through the class - they usually train for a number of years, go to college elsewhere, and then we don't see them for a while.  A few times we've had parents who train with their own children, sometimes multiple children, which is sort of cool, but not the point of this post.
A popular sport for high schoolers in that area is cross country track.  If you're not familiar with it, in cross country one runs a few miles...across country areas.  Hence the name.  I think the events are usually around 5km, which is a bit over 3 miles.  The point is, they're not sprints, they're distance events.
We've had a few of our students joint their high school cross country team in order to "get in shape."  That follows the conventional wisdom pretty well - after all, nothing gets you healthier or in better shape than running long distances, preferably while indulging in a high carbohydrate, vegan diet.  Right?  (Please note the sarcasm here).
In a similar vein, when adults I know want to get in "better shape" for, perhaps, a promotion (belt tests can be grueling in my style), they take up jogging.  You know - get out and run 3-5 miles at a time.  Again, this matches the conventional wisdom nicely.
The problem is that running distances (anything over a mile or two) is probably not the best way to get "in shape" for karate.  In fact, it may be more than inefficient - it may be counterproductive.  Why?
The problem is specificity.  Your body adapts pretty specifically to the demands you put on it.  If you do a ton of wrist curls, your wrists get stronger, but your legs don't.  If you sprint, you get faster and better at sprinting, but that won't allow you to run a fast marathon.  Running long distances at a relatively constant pace makes you good at moving: 1) relatively slowly (the steps in a 5km race are nowhere near as fast as your steps during kumite should be), for 2) a long, steady duration, with 3) relatively little force.  It will hike up the efficiency of your aerobic energy pathways (which aren't the ones used in karate).  It will also make your body lean towards developiong slow twitch muscle fibers (the ratio of fast twitch to slow twitch fibers is partially genetic but there are undecided muscle fibers, especially in kids, that can go either way depending on your training).
Being a good karateka requires the ability to generate large forces (big force = big acceleration = speed) over short bursts of time (watch people spar - it's all stop and go, it's not a constant long duration output of energy) using anaerobic energy pathways (you just can't generate maximum force with aerobic energy) and fast twitch muscle fibers (the kind that fatigue quickly but produce lots of force).  The same is true of kata.  Explosions of movement, pauses, etc. 
Notice anything?  They don't match.  The unfortunate fact is that building up a large aerobic endurance will do little or nothing to enhance your anaerobic work capacity - the ability to do lots of anaerobic work in a short period of time without getting gassed.  Building up to running 5, 7-minute miles in a row will not make you better at sparring all-out for 2 minute rounds, or at least not much better.
What's wrong with running distances?  Well, here's a few problems:

1. High injury rate.  Runners tend to get hurt, unless they're freaks like my friend Kathy, who can run a marathon before lunch, then eat and run another one and never get hurt.  But she's a freak, and you're probably not.
2. Repetitive motion over a limited range.  Runner's joints take a pounding - repeating the same kind of impact over and over, for thousands of reps, can permanently damage your body.  Watch the actual range of motion in a distance runner - the angles generated at the hip and knee.  They're pretty small.  Do you think that's conducive to your ability to kick high?  Remember, your body only wants to maintain the capacity it needs.  Convince it that it only needs thirty degrees of hip flexion and that's all it will maintain.
3. Conversion of fast twitch to slow twitch muscle fibers.  Basically, it makes you slow.
4. Development of the wrong energy system.  Surprise!  Begin able to jog forever doesn't help your endurance at higher intensity levels.

I have seen high school kids run cross country, then at season's end come back to class.  They think they're in great shape, but they're huffing and puffing after a few rounds of jiyu kumite (free fighting).  The big benefit is that it makes old geezers like me seem faster and in better shape than we are.
Why do I want you to combine karate practice with cross country and/or jogging?  Because if I ever spar with you I want every advantage I can get.  I want you slow, out of breath, and nursing some nagging injury or movement restriction.  If you've ever seen me fight you'd know I need every advantage I can get (I'm not what you'd call a natural athlete - in fact, pretty much the opposite!)
If you want to be good at karate, however, focus on shorter bursts of higher intensity exercise.  Sprint 100 m (the length of a football field, more or less).  Really sprint it - go all out, knees high, so at the end of it you're totally dead.  Walk slowly back to where you started and do it again.  Repeat 5-10 times, if you can.  That's teaching your body to put out a lot of force and energy in a short period of time, which is exactly what you need to move and throw a strong punch or kick.