Generally speaking, most of my research (years ago) into the tumbling/rolling mechanism illustrated that it is a simple recessive factor in its mundane form, but that the "degree of performance" (frequency, number of revolutions, type of revolutions, rate and depth of descent, etc) is a matter of genetic selection controlled by a very wide range of different genes. To make matters worse, the tumbling/rolling mechanism is also heavily influenced by numerous "environmental" factors as well. Something as simple as regulating the amount of water or feed a roller intakes before flight affects the performance, as does the amount of light. So too can the area in which they are flown if the conditions has some affect on their flight. Treed areas for example, tend to cause them to race around assorted obstacles and this is not a flying behavior conducive to performing. Save for rolldowns or other birds that are more or less genetically possessed by this genetic mechanism, most tumblers and rollers can shut off their performance. In addition to being genetic, it is a somewhat co-voluntary thing in that while the birds cannot completely shut off their genetic disposition, they can certainly learn that their behavior has an impact. Some rolldowns or other types of excessive performers often learn to develop methods of self preservation, like developing a refusal to fly or developing peculiar habits off racing around obstacles at high speeds. Both of these behaviors seem to originate from the bird developing an inherent fear of the roll.
In otherwords, this is a very complex subject and even after 35 or so years of studying this family of breeds, I am convinced that I have more questions about the subject than when I actually started.
That being said, it is not uncommon to see F1 roller/tumbler crosses perform to some degree, due largely to the fact that a large percentage of breeds were actually derived from a tumbler background. Racing Homers are a perfect example. While typically viewed as the closest breed to the Wild Type (ie. the original Rock Dove/Columba livia), Racing Homers actually have some tumbler ancestry. It is generally acknowledged that the Cumulet was one of the parent breeds of modern Racing Homers. Cumulets are technically highfliers, but this places them within the tumbler family. Fantails, as far removed in type as they are, are also descendants of the tumbler family. Feral Pigeons too, contain large amounts of tumbler/roller blood, and in some areas where Rollers are kept (and lost) in very large numbers, it can make up a majority of their ancestry.
Putting it another way, the results of tumbler/roller crosses really has much to do with the breeds involved and often the outcome is actually fairly predictable.
When I was doing research on this subject (strictly for my own use and amusement), I personally found that the best cross to study tumbling/rolling genetics was Birmingham Roller X German Toy/Field breeds, as the latter seem to actually be the only group of breeds that has not been heavily influenced by the Tumbler family as a whole. Crosses into Archangels, Suabians, Monks, Swallows, Larks and other breeds tends to stomp out the tumbling/rolling completely, but breeding these F1 crosses together produces close to about 25% tumblers.
White King X Roller is not a cross I ever personally performed, but the genetic background of the White King gives us some insight into the probable outcome. It is generally acknowledged that the first White Kings appeared in the 1890's and arose by crossing White Mondains, Runts, Maltese and Squabbing Homers. While Silver Kings were developed independently some years later, the breed combination was not much different. In the hundred years since, both these genepools have been crossed back and forth to the point that they are no longer independent of each other. At the same time, it is usually thought that White Schietti Modenas were crossed in to produce Show Kings. And in the meantime, a lot of crossbreeding has taken place between Show Kings and the original Utility Kings.
Taking all this into consideration, there is very little likelihood that your King has any tumbler background. This would ultimately mean that your King X Roller cross will probably not ever exhibit any indication of tumbling. I would tend to agree with Sourland that the closest behavior that you might see is some very mundane tail riding.
But I doubt it.
