Good question. So,
Firstly you have to understand that (from the Wiki)
The "tribute system" as a term is strictly speaking, a Western invention. There was no equivalent term in the Chinese lexicon to describe what would be considered the "tribute system" today, nor was it envisioned as an institution or system. John King Fairbank created the "tribute system" theory in postwar United States to describe "a set of ideas and practices developed and perpetuated by the rulers of China over many centuries."
"So-called" "Tributary system of China" was, actually the trade mission that benefited both China and the country who made a "tribute". As the same Wiki says,
The tribute system also functioned as international trade with China and could be profitable to tributary members, as was the case with Korea. Thus, there was an economic incentive to be a member of the tributary system.
So your question,
The usual picture I have of tribute to the Chinese emperor, is that tributaries were dispatched to present their tribute to the court, that is in China.
would be, I am sorry, off the point, since there is even a Wiki list of countries that received the "re-tributers" (my own word).
According to the list, there was a country that received the "re-tributers", from the Wiki,
Xiongnu in 200 BCE-138 BCE: the Xiongnu repelled the invading army of the Western Han Dynasty, advanced into the territory of China, and besieged its capital. The Chinese Emperor recognized the Great Wall as the border of the two states and was obliged to pay annual tribute (silk, liquor, rice) to the Xiongnu.1
So it dates back to long before Tang dynasty, and Japan was under the Tributary system of China for long time, through at least as far as I know from the beginning of the 7th century to the late 18th century, as the Wiki says,
Membership in the tributary system was a prerequisite for any economic exchange with China; in exiting the system, Japan relinquished its trade relationship with China. Under the rule of the Wanli Emperor, Ming China quickly interpreted the Japanese invasions of Korea (1592–1598) as a challenge and threat to the Imperial Chinese tributary system. 
So the so-called "Tributary system" was based on the economic exchange, based on the power of China, which benefited both countries under the system.
I think even the Wiki information is very weak (probably the language barrier is too high).