Let\'s say that I make an image for an OS that uses a kernel of version 10. What behavior does Docker exhibit if I run a container for that image on a host OS running a kern
Docker's behaviour is no different: it doesn't concern itself (directly) with the behaviour of the containerized process. What Docker does do is set up various parameters (root filesystem, other mounts, network interfaces and configuration, separate namespaces or restrictions on what PIDs can be seen, etc.) for the process that let you consider it a "container," and then it just runs the initial process in that environment.
The specific software inside the container may or may not work with your host operating system's kernel. Using a kernel older than the software was built for is not infrequently problematic; more often it's safe to run older software on a newer kernel.
More often, but not always. On a host with kernel 4.19 (e.g. Ubuntu 18.04) try docker run centos:6 bash. You'll find it segfaults (exit code 139) because that old build of bash does something that greatly displeases the newer kernel. (On a 4.9 or lower kernel, docker run centos:6 bash will work fine.) However, docker run centos:6 ls will not die in the same way because that program is not dependent on particular kernel facilities that have changed (at least, not when run with no arguments).