These are the steps for transformation that may answer your questions:

1. Cubemap (source) - 360° videos are often sourced from six different images stitched together.

2. Equirect (video) - the cubemap is transformed to "equirect" trivially with a fragment shader—which maps every output pixel to some input pixel.

3. Perspective (output) - Mapping equirect video to a perspective projection is also done with a fragment shader, just using a different transformation depending on the focal point of the user.

4. Pannini (alternative output) - You mentioned this projection, and it's an alternative to the "perspective" projection that allows a wider output FOV that minimizes distortion in the periphery.