For now to send any requests you need request's body to be 'static: handshake

I think this restriction is unnecessary for many async runtimes that allow borrowing external environment. My proposal is to remove this restriction, allowing types such as Full<&'a [u8]> to be used as a body. Besides, it may simplify the public API a little bit.

This can be done for http1. But the implementation of the http2 requires a downcasting of body's Data, so it must be 'static. Anyway, even <B as Body>::Data: 'static is less restrictive than B: Body + 'static, it allows to use some borrowed body that produces static chunks of data. I'm not sure if it's possible to relax this restriction too. Maybe something can be invented in the future