Brain-Flak, 122 bytes

(([{}]{()<({}[()]<({}<>)<>>)>}{})<(({}({}))([{}]{})<{({}()<<>({}<>)>)}{}>)>())<>({}<<>{({}[()]<({}<>)<>>)}{}>){<>({}<>)}{}

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Works with all integers, 142 bytes

(([{}]{()<({}[()]<({}<>)<>>)>}{})<(({}({}))[({}[{}])]<{({}[()]<<>({}<>)>)}{}>)>())<>({}<<>{({}[()]<({}<>)<>>)}{}>)<>([]){({}[()]<({}<>)<>>)}<>

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Brain-Flak, 122 bytes

(([{}]{()<({}[()]<({}<>)<>>)>}{})<(({}({}))([{}]{})<{({}()<<>({}<>)>)}{}>)>())<>({}<<>{({}[()]<({}<>)<>>)}{}>){<>({}<>)}{}

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Works with all integers, 138 bytes

(([{}]{()<({}[()]<({}<>)<>>)>}{})<(({}({}))([{}]{})<{({}()<<>({}<>)>)}{}>)>())<>({}<<>{({}[()]<({}<>)<>>)}{}>)<>([]){({}[()]<({}<>)<>>)}<>

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The difference here is that while the first uses

{<>({}<>)}{}

To pull all the items from the off stack onto the on stack this one has to use

<>({}){({}[()]<({}<>)<>>)}<>

The first one pulls until it hits a zero and then stops, so this will fail to produce the correct result when there is a zero before the first swapped value. The second checks the height and pulls that many times, thus it will work for any list.