1. Here's why 

By default, shadows from Spotlights that are set to Medium or High are only rendered once at the start of Movies and Clips because it is quite taxing for your hardware to update the Spotlight shadows every single frame.

The result is that animated objects that are lit up by those Spotlights will leave a static shadow behind when they move around in your Scene.

This applies to Movies and Clips that are rendered via the Entire Clip and Image Sequence tabs.

If you only render single Photos, via the Current Shot tab, the Spotlight shadows will always be updated for each rendering.

1.1: To update the Spotlight shadows in every frame of your Clip or Movie, please select the relevant Spotlights and switch from Medium or High to Dynamic on the Light Properties panel:

Please note that every Spotlight with shadows set to Dynamic means that Lumion needs to render the entire Scene once more per frame.

With a lot of Spotlights in your Scene, this has the potential to increase render times substantially. You should only use the Dynamic shadow setting for Spotlights that are lighting up moving objects such as animated cars or people in your Movie.

See Also:

Knowledge Base: Why are Spotlights not casting shadows?