source code needs to be delivered to multiple streams

You could go to the Pending Changes preferences and set it to show all flows. It will show all flow targets for a workspace and a deliver all action will include all the flow targets. This may work for you but it can be confusing if there are a lot of flow targets because the Pending Changes view will get cluttered.

You could also try using a build with the post-build deliver option. A build definition can monitor a stream for changes and execute if there are new changes. You could use this to post-build deliver the changes to another stream. However, this only allows for one deliver target and is actually a component replace instead of a deliver. This may not be viable if you don't want to create multiple builds or the deliver target receives changes from elsewhere as well. You may want to script a build to do this instead using the CLI.

Regardless of what you decide to do, conflicts are a possibility so be aware of that when deciding on how to do this.

It sounds like what you want is the feature requested in Allow a stream to "share" a component from another stream. (127541) .  This would allow you to declare that all of these streams "share" the core components from the core stream, which would mean that whenever you delivered changes to the core stream, those changes would immediately be visible in all of the other streams (with no possibility of conflicts).   If this is what you have in mind, feel free to add a comment to the work item (it is currently marked as "resolved:later", so we'd need vocal support for the community to get it into plan.