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What is the right word for going from place to place with a purpose, but turns out futile? I don't think wandering or loitering would suit the weariness results from this purposeless transport.

I'd suggest the word fruitless.

Cambridge dictionary link.

If an action or attempt to do something is fruitless, it is unsuccessful or produces nothing of value:

All diplomatic attempts at a peaceful solution to the crisis have been fruitless.

Post and pillar was some sort of game or sport played in 15th c. England:

There at post and piler did she play. — English Poems of Charles of Orleans, ca. 1450

The first figurative use emerges shortly after:

Thus fro poost to pylour was he made to daunce — Assembly of Gods, ca. 1475.

Some have suggested the game was real tennis, the forerunner of the modern game, but I wonder whether it was considered seemly for women to play. At any rate, the game must have involved a lot of running from one place to the other, which, to those averse to exercise, seemed utterly pointless. This “lazy” view is how the phrase entered and remained in the language long after the original game was forgotten.

In most cases the order is reversed, though the original never disappeared. In Bleak House (serialized 1852–3), Charles Dickens uses both:

“… the man was so badgered and worried and tortured by being knocked about from post to pillar, and from pillar to post,” said Mr. George.

The idiom has been in continuous use and is found in every variety of English:

was it a litle measure that abraham and isaac so wandred, tossed from pillar to post (as we say) in perils &; feares and many afflictions, and yet contayned not the promise? — Gervase Babington, A Profitable Exposition of the Lord’s Prayer, 1588.

The whole of the General’s administration in this respect has convinced the Colonists, that no public department in New South Wales requires either experience or talent, be cause the General's favourites have been shifted from pillar to post and back again, without the least regard to such qualities. — Sydney Monitor, 22 Oct. 1831.

One major factor I feel for such high levels of stress overall, is that our lives today seem to have become busier than ever. Most people seem to be rushing against time, running from pillar to post. — Business World (India), 24 Sept. 2017.

Likewise, the town police department had been moved from pillar to post in the past few years. — Smithfield Times (VA),18 Jan. 2012.

This is The English We Speak from BBC Learning English and I'm with Rob, who's been moved from pillar to post. That means moving from one place to another in a disorganised and chaotic way. — BBC Learning English, 28 Jan. 2019.

The idiom can be used actively, where one runs/rushes from place to place, or passively where some other force or circumstance is the agent.

If you had a purpose in going from place to place, it's likely you were searching for something or trying to accomplish something.

The first thing that comes to my mind to describe being unsuccessful is in vain:

[Merriam-Webster]
in vain

1 : to no end : without success or result

// her efforts were in vain

Note, however, that there's nothing at all wrong with the word in the question itself: futile.

But if you explicitly want a noun rather than an adjective, there's wild-goose chase:

[Merriam-Webster]

: a complicated or lengthy and usually fruitless pursuit or search

// While all this is happening, Sheldon has been on a wild-goose chase around town trying to track down Bill Gates (Leonard decided to mess with his BFF by giving him the name of the wrong hotel).

— Jessica Radloff, Glamour, "'The Big Bang Theory' Season 11 Episode 18 Recap: Penny Envisions a New Career," 29 Mar. 2018