The horses run in the blue silks of the Sheikh's innovative Dubai-based racing company Godolphin and have benefited from a concept that is simple: remove the subject from cold, wet Britain to winter in the sun. For the likes of Linford Christie, any other training regime would be intolerable, but to make it possible for his equine equivalents it has needed the wealth and resources of the United Arab Emirates' defence minister.
And the vision. One of racing's strengths is its feeling of tradition, but that very asset can also promote hidebound attitudes. The efforts of the Al-Maktoum family, the rulers of Dubai, to develop the sport in their homeland, and Sheikh Mohammed's determination to create a top-class training centre in the desert, have been regarded in Britain as something of a curiosity.
Last year Balanchine, winner of the Oaks and Irish Derby, and Cezanne (Irish Champion Stakes) paved the way in Europe. But this season the Godolphin colourbearers, with top-level wins in Hong Kong, California, Japan, Italy, France and Britain, have taken the world by storm. And with racing's global village getting smaller by the day, they are signposting the future.
There is undoubtedly some romance in the Dubai dream. Three centuries ago the Arabian horse became the progenitor of the Thoroughbred, and Sheikh Mohammed's two racing operations, Godolphin Management and the Newmarket- based original Darley Stud Management, bear the names of two of the most famed Arabian stallions of the early 18th century.
And for the Sheikh, having some of his best horses in Dubai during the winter means that he can enjoy them at close quarters. He is the practical horseman of his family; his brother Hamdan (finance and industry minister) has a deep academic knowledge of the Stud Book and breeding, but the Sheikh, who has ridden all his life and won endurance races, is a "hands-on" man.
However, there is more to the Godolphin project, now in its second full year, than indulgence. There is essential business sense behind it. The wealth of the Emirates is built on oil, but Dubai's share has a limited lifespan and so the Maktoums must develop other aspects of their country to ensure its economic survival. Tourism is being heavily promoted, and the long-term plan is for the Arab sheikhdom to take over many of the international trading and sporting roles now held by Hong Kong, and the expansion of racing is just one step along that road. There is already one racecourse (though not yet racing) of international standard in Dubai, claimed, like the championship golf course next to it, from the implacable desert.
The green light came three years ago when the Emirates were declared clear of the deadly disease African Horse Sickness and the way was open for two-way traffic of bloodstock. Mohammed picked two of his brightest Darley Stud talents, John Leat and the former journalist Simon Crisford, to set up Godolphin and installed one of his countrymen as holder of the training licence. Then it was Hilal Ibrahim, now it is the Dubaian ex-policeman Saeed Bin Suroor; but it is his assistant Jeremy Noseda, formerly with John Gosden and John Dunlop, who is more crucial to success. The end-of-season decision about which horses will leave Europe, where Mohammed and his brothers employ nearly 30 trainers, is a team one.
In Dubai, they are based at Al Quoz Stables, a custom-built training establishment not far from the palace and camel racing centre. The set- up lacks nothing - not even a "portamosque" for the Pakistani grooms, and a majlis tent where the Sheikh holds court - and is a practical horseman's dream; what everyone would do if money was no object. The horses' food is prepared daily at their owner's private mill, their routine involves leaving their air-conditioned boxes to exercise or work, under top-class riders, in the cool of the early morning on a private training track made of special hoof-friendly sand, before being led round by their devoted "hot-walkers" to relax. Visit to swimming pool optional.
The Middle Eastern summer is too hot for horses, and the squad, which this year includes two-year-olds broken at Al Quoz as yearlings, has now returned to Britain for the European season. The winter climate in Dubai - warm and dry - means that they can be prepared in consistent conditions for early-season races. In April Red Bishop took the pounds 327,000 Queen Elizabeth II Cup in Hong Kong and the $200,000 San Juan Capistrano Handicap in California, and on one memorable day in May Flagbird annexed the Premio Presidente della Repubblica in Rome, Heart Lake the Yasuda Kinen at Fuchu, and Vettori, running for the first time since October, the French 2,000 Guineas at Longchamp, netting pounds 850,000 prize money between them and in the case of the last two colts considerably boosting their value as potential stallions.
Whether the winter regime actually improves a horse has yet to be proved but there seems little doubt that the Godolphins have an advantage in condition over their rivals until at least the middle of the year. What such an advantage is worth is impossible to quantify, but even a head can make a difference.
The project has caused discomfiture among the Sheikh's trainers in Britain; some have had their best prospects taken away at the end of their two-year-old careers. Henry Cecil, for instance, handled Vettori, Moonshell (third in the 1,000 Guineas on her return) and Classic Cliche, although some horses, like St Leger winner Moonax, have returned to their original trainers. Mohammed, leading owner here for nine of the last 10 years with horses carrying his familiar maroon-and-white livery, has reassured them that they will not lack his support in the future but, with a few well-chosen words earlier this year, reminded them they are the pipers.