Question: My body leaves; my core can stay,  

Where I shall breathe another way.  

The route of south I always linger.  

My arms have arms; my ring no finger.

What am I?

Answer: 
Are you a:

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> tree?
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My body leaves; my core can stay,

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> A double meaning: The body is leafs/leaves, and when they go away for the winter, the trunk/core remains
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Where I shall breathe another way.

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>  With no leaves, a tree can still exchange oxygen and CO2, just in a more dormant state
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The route of south I always linger.

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>  Route/root is another double meaning. [Via OP: roots stretch downward, and are harder to move the deeper they go] 
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My arms have arms; my ring no finger.

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>  The arms are branches, which can have more branches stemming from them. Tree trunks can have rings, but do not have fingers.
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Partial answer.

Are you:

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>  A planet or galaxy?
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My rough thinking:

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>  When a big planet explodes, it leaves its core behind. Also, planets have rings...
>  A galaxy have arms :) I have no idea how the other hints apply to this.
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