Synonyms for “passing the buck”?

shuffle sth off

to get rid of something unwanted, usually by giving it to someone else:

Ex. "The local authority may then try to shuffle these responsibilities off onto another authority." [emphasis mine]

Cambridge Online Dictionary

A phrasal verb, and considered "literary"**, it fits the intended usage well.

A comes to B with a problem. B is lazy and doesn’t feel like putting in the work to solve it, so they shuffled it off (on)to another department.

In some circles, it has higher register than 'pass the buck", but it might also be a "cross-the-pond" thing.

How about delegate?

to give a particular job, duty, right, etc. to someone else so that they do it for you:

• As a boss you have to delegate (responsibilities to your staff).
• Authority to make financial decisions has been delegated to a special committee.

[Cambridge English Dictionary]

tergiversate: v. to evade, to equivocate using subterfuge; to obfuscate in a deliberate manner.

scrimshank: v. (Britain, military, slang) To shirk duty.

bludge: v. (Australia, New Zealand, slang) To not earn one's keep, to live off someone else or off welfare when one could be working.

(Australia, New Zealand, slang) To avoid one's responsibilities; to leave it to others to perform duties that one is expected to perform.

(Australia, New Zealand, slang) To do nothing, to be idle, especially when there is work to be done.

If A refers the problem to B, who in turn refers the problem to C, then B is subdelegating to C. For instance, definition 2 for "subdelegate, v.," Oxford English Dictionary has the following:

2. transitive. To delegate (authority, or an office or commission) to a subordinate person, group, or body; to delegate further.

1891 Spectator 21 Feb. 272/1 The ruler gets bored with the work, and delegates his power, which is again sub-delegated.

1918 America (N.Y.) 20 July 351/2 The active enforcement of censorship is subdelegated to the second deputy of police.

1996 Digest Environmental Law (Nexis) Sept. 11 Secretary Glickman properly delegated authority to an Assistant Secretary, who subdelegated authority to the Forest Service Chief.

This is a quirky usage. That said, so is the practice of delegating an already-delegated task ad nauseam. Subdelegate also neatly aligns with other forms of secondary assignment, like subcontracting, where someone contracted to do work in turn subcontracts someone else to do the work, or subletting, where someone who has signed a lease sublets the property to someone else for a span of time.