The most simple solution would be redirecting the output of rsync to a logfile.

rsync -avz /something /somwhere >> ~/rsynclog

Another way you could is if you know the rough directory you'll be syncing (ie, we'll use the directory 'movies' for example) you can use a combination of lsof and grep:

lsof | grep rsync | grep movies

lsof will list your open files, pipes the output to grep to find any opened by rsync, pipes that output to grep to find the directory/file which is open.

Here are two ways

With screen: Attach screen session to your cron job:

screen rsync --progress src dst

this will allow you to re attach to the rsync jobb anytime you want to check what files its currently processing (just be sure to be the same user as the one that launched the rsync job) with

screen -x

With logging add logging to your rsync job:

rsync --log-file=/tmp/rsync-status.txt  src dst

then follow the log in real time with:

 tail -f /tmp/rsync-status.txt

As Király István suggested, I am running

rsync -ravz /Users/jkirby/Music/iTunes/* .

which gives output like

Jeffs-MBP-2:2016-08-15 jkirby$ rsync -ravz /Users/jkirby/Music/iTunes/* .
building file list ... done
Temp File 1.tmp
Temp File.tmp

From that output I can see what directory is being copied.

In the case where rsync is copying a lot of large files slowly, I monitor that directory using watch like so. This way I can see the temp file that rsync creates and I can see the size growing on the file currently being copied.

watch -n1 "~/Music/iTunes"

Monitor running process under GNU/Linux

Monitor rsync transfert on both ends

By running Linux kernel, you could query /proc pseudo filesystem. There is no need to resort to strace or lsof, (could be useful on small configuration where this tools could even not be installed).

ps -C rsync fw
  PID TTY      STAT   TIME COMMAND
 8645 ?        Ss     0:00 /usr/bin/rsync --daemon --no-detach
13763 ?        S      0:01  \_ /usr/bin/rsync --daemon --no-detach
13764 ?        S      1:30      \_ /usr/bin/rsync --daemon --no-detach

ps -C rsync fwo pid,lstart,stat,cmd
  PID                  STARTED STAT CMD
 8645 Sat Jul 10 14:34:12 2021 Ss   /usr/bin/rsync --daemon --no-detach
13763 Sat Jul 10 15:51:23 2021 S     \_ /usr/bin/rsync --daemon --no-detach
13764 Sat Jul 10 15:51:23 2021 S         \_ /usr/bin/rsync --daemon --no-detach

The 1st process is the daemon, looking for start time show two process started by remote rsync.

So, looking for the last one:

mapfile -t allrsyncpid < <(ps -C rsync fho pid)
rsyncpid=$((${allrsyncpid[@]: -1}))

Then with this $rsyncpid variable:

ls -l /proc/$rsyncpid/fd
total 0
lr-x------ 1 root root 64 jui 10 15:51 0 -> /dev/null
lrwx------ 1 root root 64 jui 10 15:51 1 -> socket:[106279330]
lrwx------ 1 root root 64 jui 10 15:51 2 -> socket:[106279330]
lrwx------ 1 root root 64 jui 10 15:51 3 -> socket:[106279331]
lrwx------ 1 root root 64 jui 10 15:51 4 -> /path/to/some/file.IxOWxh
lrwx------ 1 root root 64 jui 10 15:51 5 -> socket:[106323027]
lrwx------ 1 root root 64 jui 10 15:51 6 -> socket:[106320690]

Then

ls -l $(readlink /proc/$rsyncpid/fd/4)
-rw------- 1 root root 41024 Jun 10 15:59 /path/to/other/file.ZIjEDk

If you repeat this, you could see wich file are growing and how.

while [ -d /proc/$rsyncpid ] ;do
    ls -l $(readlink /proc/$rsyncpid/fd/4)
    read -t .5 foo && break
done
-rw------- 1 root root 0 Jul 10 16:16 /path/to/some/file.nMM3Lz
-rw------- 1 root root 0 Jul 10 16:16 /path/to/other/file.H2b61i
-rw------- 1 root root 0 Jul 10 16:16 /path/to/some/bigfile.B0Xgg7
-rw------- 1 root root 2097152 Jul 10 16:16 /path/to/some/bigfile.B0Xgg7
-rw------- 1 root root 42359296 Jul 10 16:16 /path/to/some/bigfile.B0Xgg7

Until you press return or the end of rsync process.

Of course, as you are watching for /proc entries, created and deleted very quickly by another process, there are some chance ls is executed too late after readlink. So this kind of message is not harmful:

ls: cannot access '/path/to/some/file.fKSZss': No such file or directory