Calculation of the integral image is a linear transform on the original image.
Now, assuming you wanting to downscale the integral image means that you want the result to be the same whether you:
- downscale the original image, integrate after
- integrate the original image, downscale after
and remember that "downscaling" by a factor of $N$ the original image means
- applying a low-pass filter to avoid aliasing and
- throwing away $N-1$ of every $N$ pixels.
since low-pass filtering is also a linear operation, we can just switch orders of summations and down-scale the result image. Nothing special applies.