Is the question/greeting "How're you going?" correct usage?

All right. Turns out, it is a greeting, after all! Perhaps quintessential AusE (Australia & New Zealand).

Jim Nicolson, Long Creek, 2010, p.145 (GoogleBooks) (emphasis added)

(From the Preface): The principal settings are two cattle runs (ranches) in Australia's Northern Territory … In 1947, Darwin's population was 2,538 …
…
"G'day, Murranji," he called out. "I thought you were still over at the River." "Yeah? How're you going, young feller?" "Good."
…
"No way Jack'd miss the races and a chance to catch up with old mates. How're you going, Jack?" Brian called out. "Not bad. An' how're yer drinkin', fellers? Left any beer in O'Hara's?"

See also:

• Ron Player, Under Different Skies, 2010 (GoogleBooks)
  (Setting: Narrow Neck Beach, Auckland, New Zealand) Note the author's remark, "That was a pretty normal greeting in those days." about "G'day, Allan, how're you going?" (p.24)

• Jan Eriksen, Goldmine Experience, 2013 (GoogleBooks)

• Di Morrissey, The Islands, 2008, p.535 (GoogleBooks)

Plus
D.H. Lawrence, The Primrose Path, in "England, My England":

‘And how’re you going on, lad?’

If you're not sure, stick to what you're certain to be correct! I've been living in England for the past 15 years and have been studying English grammar for about 14, and have not heard 'how're you going' as a way to greet someone else.

It must, most certainly, be an Australian and New Zealander expression, but I have to say though I haven't ever heard my Aussie friends saying it; perhaps because they might know that it's not the way the Brits would say it.

My suggestion is, always communicate in the same way the locals do; that shows respect,that you're well integrated in their way of living, and you'll sound less foreign.