Thanks for the feedback! In general, these features would be up to the Custom Tile developer to implement.
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Thanks for sharing the example use case with WeatherWidget.io – in that case, I assume you’re using the URL option for Custom Tiles directly. You may be interested in using the Refreshable URL Embed custom tile. The Refreshable URL Embed custom tile is like the normal URL custom tile option, but adds a setting for periodically refreshing the URL.