As someone who began an emissions-legal engine swap years ago (5.0L Fox-body Mustang V8 into a 1991 Miata), I have spoken at length with the Department of Air Quality. These are the guys who must personally inspect and test your completed swap, declare it emissions legal, and sign off on it. It would be wise to contact them in advance like I did so that you know what is required. To sum up what they told me:
1: the emissions decal of the recipient vehicle reigns supreme. Whatever it calls out will usually still be required. In other words, if the original engine had EGR then any swapped engine must also have EGR. (This is why I selected a 1.6L Miata as my recipient--that engine never had EGR. The '94+ 1.8L Miata did have EGR, and DAQ said I would have been required to have it on my V8 engine had I selected that newer model.)
2: if you are significantly updating the emissions system (say, going from a '78 carbed engine to a multi-port EFI setup) and the swapped engine runs very clean, you might be able to get a waiver for select "lesser" emission components the recipient vehicle originally needed. Example: a former co-worker pulled the original non-feedback carbed engine from his '79 Subaru Brat and replaced it with a later EFI Subaru engine. His car originally required an air pump, but the new engine ran clean without it so the DAQ granted him a waiver so it was no longer needed to maintain compliance.
3: if your donor engine has additional smog equipment that the recipient vehicle lacked (such as an air pump), you may still need to run it in order to pass a tailpipe test. If you can get the resulting project to pass the test without it, however, you should be fine.
That is a short overall summary. Like I say, for the best possible info it really is worth calling your local DAQ.