Is it possible to change an object and automatically change all copies of that object across artboards in Illustrator?

Linked Files

I use Linked files for this. To do so, save the artwork you’d like to duplicate into a separate file, then use File > Place and make sure “Link” is checked in the Place dialog. The list of Linked files in any Illustrator document can be found in the “Links” pane.

AndrewH already pointed out you can use Symbols, but I try as much as possible to avoid using Symbols. Two main advantages to using Linked files are file size and file organization.

Advantage: File Size

Illustrator does a poor job of keeping files small when using Symbols. To show this, I duplicated a the same artwork 1, 2, 4, 8, and 16 times using both (static) Symbols and Links:

Links behave as you might expect: a few bytes increase in file size for each Linked file. Symbols on the other hand do not, they added almost 10MB per duplicated symbol despite it being “static”.

Advantage: File Organization

Unfortunately, Illustrator does not allow you to create symbols if the artwork contains a linked image:

I prefer to not embed linked files into the Master AI file as much as possible, particularly with raster PSD files. If a PSD file is embedded in a document and you need to edit it, you need to first un-embed it, then make your changes, then re-embed it. This gets messy very quickly.

Disadvantage: Updates/Refresh Time

Illustrator’s detection for changes in Linked files can be sluggish. If I update a linked PSD, Illustrator might take some time to notice before it asks to update the Links. Sometimes it may not notice at all.

You can always manually do so by clicking the “Update Link” button on the Links panel, but ultimately Symbols have the advantage here. They should update right away since everything is contained within the same Illustrator document.

Precaution: Links should be well contained

As joojaa pointed out in the comments, you should be careful you’re not linking to common resources. If two AI files both use the same external file as a Link, then making changes to the Linked file will affect the design of both AI files. Sometimes that is what you want, other times that it not what you want.

My personal workflow is to have a separate folder for each project and keep all Linked files within that folder. The File > Package... command can come in handy here to clean up your project folder if needed.