I think that, in general, strikes have been more neglected by judo practitioners than grappling techniques have been by karate practitioners. That said, I do still advocate cross-training--the grappling found in karate, even when trained effectively, is intended for use on untrained opponents, so it is fairly low-level grappling. If you want to have enough grappling to handle yourself against the average tough guy, you can find karate instructors who know enough. If you want to be as effective as you can possibly be then, of course, you need to cross-train in a grappling art. Even karate masters of old advocated this, many of them regularly pursuing tegumi, sumo, judo, aikido or jujutsu. I, personally, spent 4 of my 6.5 years (so far) of karate training also training in judo, and I constantly incorporate it into my karate. I never was a great judo competitor, but when combining it with strikes and controlling techniques of karate, it works fantastically.
For those into the "found-grappling-in-kata" thing, many bunkai will make a lot more sense as nage than as strikes.
However, you will only figure this out in a physical sense if:
1) You already had a grappling background before taking up Karate
2) You cross-train regularly in some form of grappling
3) You practise those bunkai in an alive manner
4) You have a job where you have to put your skills to use.
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It will take longer than a year to become really proficient at Judo - Probably explains why people study it for years and years. Aikido won't work at all unless you're really lucky to get an instructor who has been actively involved in resistance based training - trust me when I say, there's not many of those. Even if you are lucky, Aikido waza are complex series of movements with taisabaki which is completely different to Karate-do and judo. They take a long time to become natural movements.
Then you've got the issue of compartmentalising each of the systems, (as they all have physically different mechanical principles) so that you're able to correctly internalise the disciplines and maintain the authenticity of each through their respective gradings.
I fully support cross training however, it can be as much of a hindrance as it is of benefit to someone's skill sets. I wouldn't suggest cross-training in a system which has different mechanical principles until you have a very firm grasp on your primary art - possibly approaching senior kyu to early dan grade (depending upon how much time you have to devote to your entire training.)
I disagree with ^^^^^^^^^^^^ I think taking a new system in a different aspect (grappling) where you have almost no knowledge in (like me) and not really covered in your style, it is not really much of a hindrance.
If you are doing Kyokushin Karate for stand-up, know that the grappling is non-existent. Step sparring "grappling moves" can hardly be used and your are not even allows to grab any part of your opponent in sparring. So do Judo and learn what you need for grappling without any attachment from KK.
As far as waiting to cross train I've heard many mixed opinions on this. I currently train 5 days a week but after my tournament I'm dropping to four days a week to spend a little more time at home with my wife. I was planning on 2 days kyokushin and 2 days Judo. From watching lots of YouTube on Judo as well as picking up some books I see the footwork is very very different. As far as breaking balance of your opponent, we cover that slightly when being taught street self defense which is only a couple times a week. Could you possibly explain the negatives of cross training early into my training please?
Study what is at your school. You are showing the problems some people have when preparing to cross-train. You aren't getting a "head start" by reading and viewing Judo videos. You are already starting to analyze differences with only one months in one art and zero in another.
You have a tournament, focus on that.