Argument refuting discreteness of spacetime

I have a question about following statement by Luboš Motl refuting that distances or durations could become discrete near the Planck scale:

[...] The proposition that distances or durations become discrete near the Planck scale is a scientific hypothesis and it is one that may be - and, in fact, has been - experimentally falsified. For example, these discrete theories inevitably predict that the time needed for photons to get from very distant places of the Universe to the Earth will measurably depend on the photons' energy.

Could somebody elaborate in more details the idea in the argument in 2nd sentence how to see that in such hypothetical theories the time needed for photons to pass distances would inevitably have to depend on their energy?

Generally what you are looking for are a host of ideas and theories that suggest there are lorentz symmetry violations that can potentially be observed experimentally.

One of the more speculative ideas was Double Special Relativity

Since DSR generically (though not necessarily) implies an energy-dependence of the speed of light, it has further been predicted that, if there are modifications to first order in energy over the Planck mass, this energy-dependence would be observable in high energetic photons reaching Earth from distant gamma ray bursts. Depending on whether the now energy-dependent speed of light increases or decreases with energy (a model-dependent feature), highly energetic photons would be faster or slower than the lower energetic ones