So I decided to continue in my exploration of the Power BI API. The next natural step in creating customized dashboards is the ability to dynamically filter the tiles based on user interaction. As of today, this is something you cannot achieve in https://app.powerbi.com (you can only filter using selectors in Reports).
There is a post here, with very scarce information about a parameter of the embed API, namely $filter. We can use this parameter much like its oData cousin, with the same syntax.
I tested it for string parameters and it works. Did not have such luck with dates. Everything will eventually come into place, as this API is still under development as I understand.
I created a sample web page trying to keep it very simple, to demonstrate how this works. You can find it here and you are welcome to help yourself reusing my code and adapting it to your needs. In the example you will see a list box, clicking on it will trigger a refresh of the tile with the appropriate results.