Gekisai Dai and Gekisai Sho were developed from Sosai Masutatsu Oyama’s training under Nei-Chu So sensei short after the WO II. Master So was a top student of Chojun Miyagi sensei, founder of the Gōjū-ryū 剛柔流, one of the main traditional Okinawan styles of Karate, featuring a combination of hard and soft techniques. Chojun Miyagi developed Gōjū-ryū from the Naha-Te - 那覇手 system of Okinawan Karate, which originated from Southern Chinese Kempo.
Gekisai Kata were orginated in Okinawa by Soshin Nagamine and Chojun Miyagi sensei from Fukyu Kata Ichi and Fukyu Kata Ni. Chojun Miyagi adopted Fukyu Kata Ni as Gekisai Dai Ichi and went on to develop Gekisai Dai Ni with its Naha-Te influence as a training Kata into the Gōjū-ryū curriculum in 1940 for school children and adolescents.
Essentially, the two Fukyu Kata were Pinan equivalents in that they were designed to be summation of the Karate that went before. They are new Kata (made in the 1940s). The idea was to create standardised Kata that would cut across all the various streams of Karate, that were suitable for novices, and would provide a common grounding in the basics of Karate. The Pinans were considered to be a summation of Shuri-Te line alone (quite rightly) and hence were lacking the Naha-Te side of things.
The name Fukyu - 普及 translates in something like 'universal', 'popular' or 'widely spread'. So the name of the Kata matches the intention behind their creation. The revised name of Gekisai - 撃砕 translates as 'Pulverise' or 'Attack and Destroy' which would seem to be more reflective of the intent of the applications; as opposed to the former name which reflected the political intent of the Kata. Gekisai is derived from the characters Geki - 撃, meaning to 'fight, attack or 'strike', and Sai - 塞, meaning 'to smash or break'. The name reflects to the period in history when they were created, i.e. WW2 and its inference was 'attack and smash the enemy', i.e. the American soldiers. The original upper punch taught was higher than head height, reflecting the height difference between the Okinawans and the Americans. In post war years many Gōjū-ryū schools have changed these punches to standard upper and middle punches.
Gekisai Dai - 撃塞大, where Dai - 大 means 'majer' or 'larger', teaches strength through fluidity of motion, mobility and the utilization of various techniques. When performing Gekisai Dai, one should remember that flexibility of attack and response will always be superior to rigid and inflexible strength. Gekisai Kata was strongly influenced by the Shuri-te techniques from Master Anko Itosu.
Gekisai Dai is known as a Southern Kata within Kyokushin Karate, developed from Mas Oyama's training under sensei Nei-Chu So. Sensei So was a student of Gogen Yamaguchi (Goju-ryu) in Japan. Chojun Miyagi developed Goju-ryu from the system of Okinawan Karate, which originated from Southern Chinese Kempo.