[stoppingdown]

My take: noise won't change, but I think that more pixels (with the same noise level) can produce an inferior noise level on the output (assuming the output image size don't change), so you might lose something. I'm trying to explain with an example and round numbers. If you start with 8000x5300 pixels and produce an output of 4000x2650 pixels, each output pixel is the average of 2x2 captured pixels (this is rough and somewhat inaccurate, but I think you get the point). Averaging reduces the noise on the output. If you start with 4000x2650 pixels the output doesn't get any averaging. If you start with 5300x3500 (APS-C crop) you have averaging, but with a smaller window than the original.

But the best way to deal with the issue is to shoot FF and APS-C with the same framing and processing and compare the outputs.