When the DHCP server isn't available immediately (which typically
occurs after a power outage), I get several lines

IP-Config: no response after xxx secs - giving up

which is normal, but after a few minutes, I get the following error:

no search or nameservers found in /run/net-.conf /run/net-*.conf

and it is impossible to unlock the machine remotely!

These messages come from
initramfs-tools/scripts/functions:configure_networking() not
src:dropbear.

dropbear-initramfs doesn't configure the network on its own.  It calls
configure_networking().

Sounds like it should be a wishlist bug for initramfs-tools-core to make
the delay configurable or not give up  so easily.

In principle dropbear-initramfs could add some network detection checks
and call configure_networking() over and over untill the networks is up,
but that logic doesn't belong to dropbear-initramfs IMHO, hence closing
this.

As a workaround, you could add a `sleep 300` or so delay in a custom
init-premount script (make sure it starts before dropbear though).  This
might be OK as long as you don't reboot the machine too often.

BTW, using a static IP instead of DHCP is not an acceptable workaround
as it is then impossible to unlock the partition by typing the
passphrase with the keyboard! (When I type keys, nothing happens.)

Uh?  I fail to see how network configuration details impact the input
method for remote unlocking.

I've reported a wishlist bug against initramfs-tools-core
about the DHCP retries here:

https://bugs.debian.org/cgi-bin/bugreport.cgi?bug=1146486

I'm wondering. Perhaps an obscure bug somewhere.