Why some Tile Previews for External Reports load takes time (from 10 sec to 1 min) even though the Previews show pre-captured images (not live visualizations)?

Prior to v6.1.2. The Tile Previews loading may take time if they have External Filters. 

For External Reports with Filters Tile Preview images are collected and cached only when you open the Preview directly. The image that was collected during Trigger run is not cached therefore makes no impact on the picture load for Tile Preview.
The cache is cleaned after every trigger run, so for daily triggers the first Tile Preview opening will take time every day. If you open the same Tile Preview several times per day, subsequent openings load quickly.

As for External Reports without filters, Tile Preview loads the image that was collected during Trigger run as it is cached.

Starting from v6.1.2 the way it works for External Reports with filters was changed: the image collected during Trigger run is also cached with default set of filters. Tile Previews use that image so the load is faster.

The Tile Preview loading will still take time if there is a personal set of filters for a certain User:

In last two cases the Tile Preview is to be loaded personally for certain User with a personal set of filters.