The title pretty much conveys what I want to ask but I would elaborate a bit. If a question is appropriate to be closed due to either of the listed reasons for a question to be closed, should one also downvote the question provided one is voting to close it? In certain categories such as the question being belonging to a different site or the question being a duplicate, I feel it is clear that it doesn't make much sense to downvote as the question simply doesn't warrant being judged on its quality--it simply doesn't belong here or shouldn't exist as a separate question--so up/down voting seems moot. But, what about the cases when the question can be judged for its quality as a physics question but also deserves to be closed? Is down-voting superfluous then or still an important element in the healthy functioning of the "reputation economy"?
The main advantage of downvoting that closing does not have is that it quickly removes questions from the front page, once the post has a score of -3. If you think a question is probably unsalvageable, by all means downvote it.
If you are voting to close an otherwise good question as a duplicate, or if you think that with some care the question could be reopened, then you shouldn't downvote, since removing it from the front page will only make it harder for it to receive said care.
That said, the only rules on voting are not to serial vote or game the system, so you can use any system that makes sense to you.
I downvote to say "this content shouldn't be here", and with the exception of some duplicates, that's what the approach to close-vote-worthy questions should be. Downvotes send yet another message to the OP that the content has a problem.. Furthermore, if a lot of people downvote, the post gets moved off the front page, which helps keep it a bit cleaner.
Regarding duplicates: if a post is extremely similar to a suggested duplicate (or if the duplicate is suggested in the "related" bar), I will usually downvote, since those are cases where the OP clearly didn't look at what they were shown. But if the duplicate is found somewhere deeper, then it's nice to keep the question which I'm voting to close as a signpost for future readers who may use certain search terms or phrases, and in that case, it could even use an upvote if it's interestingly written.