Czech transport API.
Transport for Czech Republic API is a free Transportation API that requires no API key, is served over HTTPS and has unconfirmed CORS support.

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Deutsche Bahn (DB) API
Yes. The Transport for Czech Republic API is listed in the Transportation category of the public-apis directory, which only includes APIs that are free or have a free tier. Check its documentation for rate limits and paid plans.
No. The Transport for Czech Republic API requires no authentication, so you can start making requests without signing up.
CORS support for the Transport for Czech Republic API is unknown. Test a request from the browser, and fall back to a server-side call if it is blocked.
Yes. The Transport for Czech Republic API is served over HTTPS, so it's safe to use in production.