The easiest way to get what you want is to use a camera with the feature built-in. Most tasks tend to be more cumbersome and time-consuming to do in post-processing than the same tasks in-camera.

If dpreview's description of Dynamic Pixel Shift is accurate, you should be able to take four hand-held shots, align, resize, and average them to produce equivalent results.

When using Dynamic Pixel Shift, image stabilization is switched on by default (a giveaway that it isn't doing single pixel shifts like its tripod-mounted counterpart) and the camera relies on the natural movement or shake of the photographer for four slightly different images. Stabilization remains on to ensure there is not too much of a framing difference between shots. The four files are then aligned and combined to produce one 'super resolution' image with increased detail, dynamic range and lower noise. Without getting into too much detail, sampling a scene multiple times with slight shifts to the image allows details to be localized with sub-pixel precision (since shifts are unlikely to be perfect multiples of one pixel).

I would use align_image_stack and ImageMagick. Here's how to create a super resolution photo with any camera describes using Photoshop. It also seems that using many more than four images is unnecessary.

Interestingly, the difference between the 4 image and 20 image super resolution examples is less noticeable. While there is some advantage to stacking more images, returns are diminishing in this case.

See A Look at Reducing Noise in Photographs Using Median Blending.