I had a similar requirement and finally settled on:
kubectl exec $TARGET_POD -n MYNAMESPACE -- tar -zcvf - -C /my/src/directory . | tar -zxvf - -C /my/dest/dir
This depends on 'tar' being available inside the container. Similar can be done with any archive command (zip/unzip or cpio) that can deal with stdin/stdout as long as the command is in both the container and the host.
Also note that '*' is not native to tar, but depends on shell expansion (in my experience...), so if you want to match only certain files in a directory on the container I include 'bash' in the call and cd to the directory first:
kubectl exec $TARGET_POD -n MYNAMESPACE -- bash -c "cd /my/src/directory;tar -zcvf - *.log" | tar -zxvf - -C /my/dest/dir
Again, bash or some other shell supporting * expansion must be present in the container.
Finally, this works in the other direction as well sending multiple files to a container:
tar -zcvf - *.log | kubectl exec $TARGET_POD -n MYNAMESPACE --stdin -- tar -zxvf - -C /my/dest/dir
And "bash" is your friend if the target directory does not exist:
tar -zcvf - *.log | kubectl exec $TARGET_POD -n MYNAMESPACE --stdin -- bash -c "mkdir -p /my/dest/dir;tar -zxvf - -C /my/dest/dir"