Connecting to my own CMS (API)?

Here are some examples, which can be easily searched on this forum.

https://discourse.webflow.com/search?q=%22cms%20slider%22%20order%3Aviews

See Creating a Dynamic Slider in Webflow - #89 by jbleroux, which uses cycle2 slider plugin. Basically any/most JavaScript slider plugin can work, you just need some JavaScript knowledge to set up, or contact a Webflow Expert.

You already can apply a filter in the collection list with date time fields. Therefore the “expired” ones won’t show.