No - Its not changing the ISO to over-expose the image; its changing the ISO to give it range to under-expose the image
Dynamic Range settings above 100 don't kick in until ISO400. So, if you set it at Auto then you either set your ISO below 400, or the camera assesses that the ISO is best below 400, essentially nothing happens (the DR setting will be 100).
If you set both DR and ISO to auto and the camera assesses that the scene will have blown highlights, it will set the minimum ISO possible to initiate DR200, under-expose the image by 1 stop and raise the shadows by 1 stop. Depending on the mode you're in (Aperture or Shutter Priority) it will adjust those settings to maintain its ideal exposure.
If you
force the DR setting to either 200 or 400, and your camera is set to automatic for ISO, then it changes the ISO to the lowest setting that the DR setting will allow, to give it the range to under-expose the photo and then bump up the shadows by 1 or 2 stops.
Now - if you force DR to 200 or 400, and then configure your ISO - the camera will always respect the exposure triangle before it applies other processing, so if you set your camera manually to DR400 and ISO 125; you're actually getting DR100. The camera will illustrate this by turning the DR icon yellow.
Simples
PS: The minimum ISO could well be different depending on the camera. I think for DR to kick in on my X100F, the ISO has to be above 400. On my X-T5, I think its 500. Which would make sense, as the newer, 40MP sensor probably inherently has more dynamic range before it needs to start applying processing.