Ashihara split from Kyokushin 1980, The founder of enshin followed ashihara, but later split with ashihara aswell.
World Oyama was founded by Shigeru Oyama (no blood relation to the founder of Kyokushin) who left kyokushin 1980 (+/- a year or so, I cant remember).
Steve Arneil left kyokushin 1991 and founded his own kyokushin organisation called IFK. It is now quite large and well respected.
Nakamura split with kyokushin 1976 and founded seido juju. Mr Ichi split with kyokushin 1980 with ashihara, but split from ashihara a few months later to found the seidokaikan (aka shodokaikan) group (and also originaly founded the K1 tournament).
Yoshiji Soeno left kyokushin 1981 to found shidokan.
Jon Bluming was expelled from kyokushin 1967. he later started a organization that after several namechanges now call themself kyokushin budokai. (and teach a mix of kyokushin, kickboxing and judo.
When Masutatsu Oyama (founder of kyokushin) died 1994, there was a squable about his will, and who was supposed to inherit the organisation IKO. There has been several long legal processes, and words as "forgery","yakuza" and similar has been used.
Anyway the original IKO is now split into 4 groups called simply Iko-1, iko-2(now also known as wko in japan), iko-3 and iko-4 to keep them apart, and there are 2 major other factions (kyokushin union and kyokushin-kan/Kyokushin Shogakukai) and several minor ones.
The kyokushin organisations all still has the same rules. Only difference I have seen is that some differs 2 seconds (from 3 seconds to 5 seconds) inorder to get up after a knockdown or it is scored a full win instead of a half win. No grabbing, even for a instant, is allowed nowdays.
I think world oyama is the same.
Enshin and ashihara allows a brief grab for a throw or pull off ballance.
Shidokan used to allow grabs and throws with limited ground grappling, but now divides its competition into 3 parts (1 pure knockdown, one muaithai and finaly a grappling part).
I Dont know what the seidokaikan/shodokaikan knockdown rules are, but they still do knockdown internaly, despite their involvement in k1 kickboxing.
The basic knockdown karate rules:
No protective gear allowed, exept a mouthpiece (optional) and groin guard (required).
Kicks are allowed anywhere exept groin and joints (read "knees") leg kicks are very popular, and punches are allowed everywhere exept the head and groin. Elbows are allowed,but not against the head, so they are seldom seen.
Spine is also a prohibited target area.
Winner is first to a full point. a wazari or half a point is awarded if you knock your opponent down and he gets up within 3 second (or 5 depending on organization), if it takes longer (or if he is unable to get back up) a ippon or full win is awarded.
If no point is given or both stands at one half point each, the 5 judges may give the win due to obvious superiority, but the probable result will be a extention (judges tend to want a win to be by ippon, and gladly gives extentions).
In the case of a extention any half points are removed and both fighters start from scratch.
After the 3rd extention (each round being 3 minute and extentions being either 3 or 2 minutes depending on organization) the opponents will check their weight, and if there is a 10 kg (again might differ slightly depending on organization) difference the lighter fighter will win. otherwise there will be a 4rth and final extention where the judges MUST decide on a winner.
Also before the semifinals there will be a tamashiwara section, where everyone will be required to break a minimum nuber of planks or forefit the tournament. You MAY request to break more however, and in the case of a draw later on, you can be given a victory if you broke more planks than your opponent. But Im not entirely sure about how that works.
Knockdown fights tend to be brutal and in-elegant affairs. The parody of a knockdown bout is two large guys, resting their foreheads against eachother, trading punches and lowkicks, both refusing to block or move anywhere but forwards (moving backward might be seen as a sign by the judges that the other is "superior"). This was the way knockdown used to be, and unfortunaly still occationaly are. But it is seen less and less frequent, as footwork and sabaki (that is found in kyokushin aswell as in ashihara and enshin. only we are not quite as fanatic about it as they are) becomes more and more prominent.
