How to say it: Fleeing from responsibility

"He flees from responsibility" is correct but not common.

A more customary word is shirk.

Definition:

Avoid or neglect (a duty or responsibility).

Example:

Their sole motive is to shirk responsibility and rip off the company.

(Oxford Dictionaries Online)

Slack also slack off

To make less effort than usual, or to be lazy in your work. Longman

To be careless or remiss in doing. E.g. Slack one's duty. The Free Dictionary

To avoid work. Merriam Webster

An informal phrase much in use in places I have worked in the past is he has slopy shoulders, the metaphor involved being the shouldering of responsibility. Since his shoulders are slopy, he cannot or does not do this.

I can't find a reference to confirm this, sadly.