Centrifugal compressors want a pretty steady suction pressure and a pretty steady discharge pressure.  That is because they'll only do about 2.5 compression ratios per stage.  Consequently I don't recommend them for wellsite applications where I would have to hold significant backpressure on the well to maintain my assurance that the suction pressure remains in the design range.

When I say that in my classes, I often get the question "but you say that centrifugal machines are the most common compressors offshore, how are offshore wells different?" and my answer is that offshore wells have to be willing to accept a higher well-abandonment pressure to be able to have the energy density required by limited deck space offshore.  That usually carries the day.

Today I was talking to a bunch of Engineers and the question came up again, and I gave my pat answer.  One guy came back with "What happens if I just let the suction fall on a 4 stage centrifugal?".  His machine is designed to go from 60 psig to 2000 psig across the skid.  If the suction pressure dropped to 5 psig then instead of doing 27 compression ratios he'd need to do 117, and for a 4 stage that would be 3.3 ratios per stage which the machine just can't do.

Does anyone have a feel for how a centrifugal would look when it saw that sort of variability?  Would it just stall out like a centrifugal pump, or would it go into surge?

centrifugal compressors generally have a fixed geometry, thus each stage/wheel will only produce a specified amount of head at a given speed.  centrifugal speed ranges are not that high due to rotor critical speeds (i.e. < 8,000rpm for example) and stresses in the wheel.  without doing the math analysis, to maintain the same inlet flow cfm at the higher ratio, the compressor speed would need to be increased.  there are power and speed limitations, not to mention rotor stability as well.  if operating conditions changed as you described (i.e. increased compression ratio), the operating point would move towards surge and most likely the anti-surge system would take control.
other operational matters to consider are gas velocities through the piping and outlet gas temperature limitations.

Depending on the exact design of the compressor the answer can vary to some extent.

If we assume that the only thing that is changing is the inlet pressure to the compressor, then the compressors capability to compensate will be a function of the wheel/impeller geometry and where it is operating on the performance curve, but we can typically safely make the following general statements.

As the Pin falls (and thus the Pout falls) a variable speed compressor's control system will attempt to continue to meet the discharge pressure required by increasing speed until max speed is reached. Once the max speed is reached that is it, the compressor is operating at its maximum pressure ratio, and your max discharge pressure will be a simple function of the stack-up of pressure ratio's and losses across the stages and coolers.

If your Pin continues to fall and you have a matched demand downstream the operating point will simply move down the curve to a lower discharge pressure and your limiting factor as pmover eluded to will be the max velocities through the gas passages internal to the compressor. (Note: As Pin falls the mass flow through the compressor falls and thus the power falls. Some surge control systems monitor absorbed power and will engage as a safety measure if it gets too low.)

If on the other hand the downstream demand is not matched (i.e. you are pumping against a check valve that has closed) the antisurge system will engage according to delta P and the associated control line (i.e. the recycle valve will open). It is important to note that most centrifugal process gas compressors  are commissioned under these conditions, and only after full gas pressure is applied to the inlet will the compressor generate enough pressure to crack the check valve.

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