Wireless doorbell systems that have a kinetic-powered button (i.e. no battery) exist, e.g. https://www.amazon.com/gp/product/B0929729L1
Wireless relay receivers also exist, e.g. https://www.amazon.com/gp/product/B0B125VLNB
These products often use the 433 MHz band. Are there any that actually work together? What I really want is a kinetic-powered doorbell button that triggers a relay instead of a chime box.
You need an EV1527 compatible transmitter (the receiver linked is the common EV1527 type).
Like this one https://qiachip.com/products/qiachip-wireless-doorbell-no-battery-required
Or this https://wiki.dfrobot.com/SKU_TEL0146_Self-Powered_Wireless_Switch_433Mhz