How to strace all write calls (to stdout/stderr) from all child processes inside a docker container?

I'm running this command:

strace -f -e trace=desc ./my-program 

(So you see I'm running the -f parameter)

Which lets me see the stdout/stderr write commands of the parent process:

[pid 10] 07:36:46.668931 write(2, "..\n"..., 454 <unfinished ...> <stdout of ..> <stdout other output - but I don't see the write commands - so probably from a child process> [pid 10] 07:36:46.669684 write(2, "My final output\n", 24 <unfinished ...> <stdout of My final output> 

What I want to see is the other write commands.

This is running in a docker container being executed with:

docker run --privileged=true my-label/my-container 

My question is: How to strace all write calls (to stdout/stderr) from all child processes inside a docker container?

EDIT: Note that this application starts and stops in about 2 seconds. strace can follow its application cycle. But tools that expect it to hang around like a server process won't work. (Perhaps sysdig can be used for this - but I haven't seen an example of this model yet).