What is the post-quantum cryptography alternative to Diffie-Hellman?
Answering myself...

There is now a very analogous alternative to Diffie-Hellman in post-quantum cryptography: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies

The research paper is very new, but if the results turn out to be secure, this is a very competitive key agreement scheme for post-quantum cryptography.