Is there any network broadcast call or something that can force all the connected DHCP clients to renew their addresses immediately?
And if not, why not? Surely this functionality has been desired by DHCP administrators from the date it was born. It makes changing a router address on a big network very hard without pervasive client automation.
It is not possible without letting the network down and up at the client network interface.
Theoretically, the DHCP server can send a FORCERENEW message to all connected clients. FORCERENEW is described at RFC3203. (See also https://serverfault.com/a/569869/107832)
Unfortunately, the famous ISC DHCP Server does not support it:
RFC 3315 states in section 19.4.1 reconfigure messages. This is, however, for IPv6 only. A message states that DOCSIS 3.0 "requires Reconfigure support (for DHCPv6) as specified in RFC3315"