The type of links you're referring to are what we call Direct Links; these work as purchase buttons the affiliate can place on their own site to sell the merchant's product(s) directly from there, rather than just referring buyers to the seller's site. These buttons only earn the affiliate a commission on each individual sale placed through those buttons, without setting a referral cookie to earn residual commissions on future purchases from the merchant.
Direct Links CAN work as Add to Cart buttons, allowing a buyer to add multiple items to their cart before proceeding to checkout; however, this only works for Direct Links from the same merchant -- i.e., items from different merchants cannot be added to a common cart together, same as with regular Add to Cart buttons. If you placed Direct Links from multiple merchants on the same site, a separate cart would be used for each merchant's items, which would probably confuse buyers.
If the merchant disables Direct Links for a product, any previously-issued Direct Links for that item would no longer function.