Most exchanges will keep those extra coins for them (Poloniex and Bittrex, for example), unless they say otherwise for a specific coin.
AFAIK, the only exchange that rewards some kind of stake to their users is Binance with NEO and GAS. They will take a snapshot of the amount of NEO their users have and will reward them with the equivalent amount of GAS every month.
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Probably the Exchange take them if they get any. Because you don 't own the coins (priv key) that are on there. You will need to put them in a wallet where you own the priv key and provides stakes like the official wallet of the coin you like to keep for staking.
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