How to create a new role that works site wide, and the default for new users

I’m new to roles and capabilities, but I’m trying to create a new role called “Manager” and I want it to be able to do everything an editor can do, plus a little more, but not quite as much as an Administrator.

But here’s the problem. When I do that using something like this:

add_role('manager', 'Manager', ...);
$role = get_role('manager');
$role->add_cap('...');
$role->add_cap('...');
$role->add_cap('...');

It only seems to apply to the current site I’m working on. If I create a new site and a new user, I can’t seem to give that new user the ‘Manager’ role without running that code on the new site first.

Is there a way to create a role that exists and can be accessed by new sites and users?