When you are behind an HTTP/S proxy and you have no other access to the internet, DNS requests by the browser or any app set to use the proxy go through the proxy. However, if you try to make the proxy transparent with something like redsocks, where you intercept UDP and TCP traffic with iptables and redirect it to redsocks, which then does its job with the proxy, this only works as expected with a SOCKS5 proxy. With an HTTP/S proxy, DNS queries bypass the HTTP/S proxy and try to get directly to the DNS server which fails, first as UDP and then as TCP. Whereas if direct access to the internet is available, DNS goes directly and site content goes through the proxy.

Is there software for DNS over HTTP/S that can pass DNS traffic through the HTTP/S proxy just like the browser does?