Swift traveller, bird lodged in shelter shortly before another tucked into meat (6,8)

I believe the answer is:

lemuel gulliver

I'm a little stuck... Click here to teach me more about this clue!

'swift' is the definition.
(I know that swift can be written as Lemuel Gulliver)

'traveller bird lodged in shelter shortly before another tucked into meat' is the wordplay.
I cannot really understand how this works, but
'traveller' could be 'gulliver' (I've seen this before) and 'gulliver' is located in the answer.
'bird' could be 'emu' (emu is a type of bird) and 'emu' is located in the answer.
This may be the basis of the clue (or it may be nonsense).

Can you help me to learn more?

I've seen this clue in The Telegraph.

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