The 5DIII let's you pick a minimum shutter speed in Av mode up to 1/250 s, in addition to specifying a range for Auto ISO (the 60D and 7D with v2 firmware, but not v1, can set a max for Auto ISO but not a min, and no min shutter).
The 1-series bodies let you specify a range (max and min) for aperture, shutter speed (not limited to 1/250 s), and ISO that can apy to any mode.
The 5DIII also lets you pick an auto shutter speed mode which is roughly 1/focal length. Which is a bit silly, since that puts your 400 f/2.8 at 1/500 when it's definitely hand-holdable well below 1/100. And, as others have pointed out, when you're in M mode but with auto ISO, the 5DIII won't let you apply exposure compensation to the auto ISO.
With ISO being so flexible these days, camera manufactures should really re-think the P/Av/Tv/M modes, and instead let you pick whatever combination of the three variables you care about, let the camera choose the others, and use exposure compensation to override the camera's determination of brightness. Ideally, you'd pick maximum and / or minimum values for all three. Reproducing today's Av functionality, then, would mean the same maximum and minimum values for both aperture and ISO but no limits on shutter speed.
Then again, I almost always shoot full manual...I only find autoexposure useful for when the light is changing too fast for me to adjust for, and I don't tend to shoot very often in those types of circumstances. Hardly ever, these days, in fact....
One final rant. Except insofar as high ISO values degrade always degrade image quality and therefore are always undesirable, as photographers we really don't care what ISO value the camera uses. If ISO ten hundred Brazilian was as clean as ISO 100, would you have a preference between the two? But aperture controls depth of field and shutter speed controls motion blur, both of which we're always looking to vary to artistic effect. So, ideally, for a given scene, you'd set the aperture and shutter to whatever you want, and then let the ISO fall wherever it needs to for a proper exposure. And that's
how I want my camera exposure system to work. We're getting there, but we've still got a long way to go....