Why do indoor sports burst shots vary in brightness under constant lighting?

Most probably, this is due to the lighting and its continuous pulses.

The Digital Picture has an excellent article called "Flickering Lights (Why are some of my pictures dark?)" that states:

The problem images [...] have usually been captured under fluorescent or gym/arena lighting and a short exposure was often used. Shooting action sports with a fast frame rate is most frequently the source – most of these images will appear properly exposed but an intermittent dark image glaringly appears.

The cause of this problem is usually not the camera, but the light themselves. [...] The more intermittent the light output, the more likely that your very short exposure may be timed to coincide with that light's low cycle. And the result is a dark image.

[...]

A great option is to strobe the venue. Securely mount [...] strobe lights in the venue's ceiling and fire them remotely [...]. This lighting technique promises the best-possible full-spectrum lighting.

[...]

The option that many [...] take is to simply deal with the conditions. Shoot in RAW and select an exposure that does not result in blown highlights when the lights are under full output.

The arguably best solution, however, is to get a fairly recent camera that features an anti-ficker-mode (such as the EOS 7D Mark II):

When enabled [...], Flicker Mode adjusts the shutter release timing very slightly so that the dim cycle of the lighting is avoided. In single shot mode, the shutter release lag time is matched to the light flicker cycle's maximum output. In continuous shooting mode, the shutter lag and the frame rate are both altered for peak light output capture. In my tests [...], the frame rate was reduced by 1-2 fps and shutter lag can be affected, making the camera feel slightly less responsive.