I don't know but with a random USB stick and a program
$ df | grep Vol
/dev/disk1s2 12228 20 12208 1% 512 0 100% /Volumes/Firmware
$ perl -E 'say $$; chdir "/Volumes/Firmware" or die "nope $!"; sleep 9999'
66433
We can indeed stall the eject, but before that we need some debugging of the process, here provided by DTrace run in some other terminal
$ sudo dtruss -p 66433
...
And then via what in English is called the force eject button, a few clicks and delays and warnings later the test program is still running, and the usb stick unmounted, and dtruss hasn't shown anything:
...
^C
$ lsof -p 66433 | grep cwd
perl5.26 66433 jhqdoe cwd cwd|rtd info error: No such file or directory
$
So for at least a standard unix program, nothing happened besides the mount point being removed out from under it. Next we can test writing something to the mount point, again with a standard unix program
$ cat writeslow
#!/usr/bin/env perl
use 5.14.0;
use warnings;
open( my $fh, ">", "/Volumes/Firmware/mlatu" ) or die "nope $!";
while (1) {
syswrite( $fh, "mlatu\n" ) or warn "hmm $!";
sleep 3;
}
$ perl writeslow
and elsewhere we confirm the cats are showing up (buffering may be a problem if you use some higher-level write function)
$ cat /Volumes/Firmware/mlatu
mlatu
mlatu
$
and again we force eject, and the program does notice this (but keeps on running, because it was written that way):
$ perl writeslow
hmm Input/output error at writeslow line 7.
hmm Input/output error at writeslow line 7.
so Mac OS X 10.11 (for this hardware is rather too old to run macOS) does nothing to "attempt to get processes to close open file handles" as claimed in the question, and there's no evidence "some kind of signal [is sent] to the processes that own those file handles" happens; instead the process keeps on running and if it does any sort of error checking then maybe it will fail, depending on how it was written.
At least for standard unix programs that have the standard unix cwd and use standard I/O calls; maybe Apple frameworked programs are somehow different? Let's remount the usb stick yet again and open the mlatu file with Hex Fiend.app...
$ open -a Hex\ Fiend /Volumes/Firmware/mlatu
$ lsof | grep mlatu
Hex\x20Fi 66642 jhqdoe 8r REG 1,5 216 7 /Volumes/Firmware/mlatu
$
(or instead use TextEdit or something if you don't have Hex Fiend installed) and once again do the force eject dance...
$ screencapture -w error.png
$

and now we get a different message than for standard unix programs and no option to force unmount.