Yes, it happened pretty much as you suggest, but people were crossing the oceans in every direction in search of gold, including from Australia to the US and then back. We have records of Europeans coming to California, and of Americans going to Canada and Australia in search of gold. Similarly people from around the world found their way to South Africa. Gold was the driver of global transfers of capital, technology and people-- for example workers from China, engineers from the US, capital from England . . . all come to South Africa

The man sometimes credited (along with others) for first discovering gold in Australia, Edward Hammond Hargraves, had in fact left Australia to go to California in 1849. He returns to Australia in January 1851, and together with four other men finds flecks of gold in February. He gives newspaper interviews -- and the Australian gold rush is on. (There are other versions of the story where Hargraves and crew aren't the first-- but he was well known, and he did very well out of it).

According to some contemporary sources, the Americans weren't all that welcome in Australia, which was more aggressively and effectively governed than California at the time. In the Times of London in October of 1852, we read from their correspondent in California:

A considerable emigration took place from [California] about six months ago to Australia, and a good many passengers are still taken by every ship. But it is rather curious that most of the foreigners who went to [Australia] from this are coming back. The Americans and French do not seem to be able to get a footing there, and they are all returning.

The most celebrated American miner to go to Australia wasn't after gold, though. Herbert Hoover, by training a mining engineer, went to Australia to mine zinc in the early 20th century, which makes him unusual (unique?) among US Presidents in having been employed overseas in a civilian capacity.

Beyond the details of the gold rushes, the comparison of experiences between the US, Canada, South Africa and Australia (there's also a very brief episode in New Zealand) is a nice platform for comparing American expansion to British colonial administration. The short form is that the American gold rushes were less regulated, those in the British territories more subject to law and policy under the administration of Gold Commissioners, an office that didn't exist in the US.

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See:

Paul, Rodman W. “‘Old Californians’ in British Gold Fields.” Huntington Library Quarterly, vol. 17, no. 2, 1954, pp. 161–172.

Hargraves, Edward "Australia and its Goldfields: a historical sketch of the Australian colonies from the earliest times to the present day with a particular account of the recent gold discoveries" [1855]

Wilson, Karen S. “Seeking America in America: The French in the California Gold Rush.” Southern California Quarterly, vol. 95, no. 2, 2013, pp. 105–140.

Gerald D. Nash, “A Veritable Revolution: The Global Economic Significance of the California Gold Rush,” in A Golden State: Mining and Economic Development in Gold Rush California, edited by James J. Rawls and Richard J. Orsi (University of California Press: 1999)

Ernest de Massey, A Frenchman in the Gold Rush: The Journal of Ernest De Massey, Argonaut of 1849,trans. Marguerite Eyer Wilbur (California Historical Society, 1929)

BAUR, JOHN E. “Californians Elsewhere: The Golden State's Nineteenth-Century Citizens At Large.” Southern California Quarterly, vol. 66, no. 2, 1984, pp. 89–132.

GOJAK, DENIS, and CAITLIN ALLEN. “The Fighting Ground Creek Quartz Roasting Pits and the Early Importation of Gold Processing Technology into Australia 1850-1860.” Australasian Historical Archaeology, vol. 18, 2000, pp. 29–38.

Higginson, John. “Privileging the Machines: American Engineers, Indentured Chinese and White Workers in South Africa's Deep-Level Gold Mines, 1902–1907.” International Review of Social History, vol. 52, no. 1, 2007, pp. 1–34.