I've been watching a BBC special on Ancient Egypt, and so far they've discussed both Howard Carter and Giovanni Belzoni.
It seems like Belzoni was barely a step above grave robbing, while during Carter's time, the transfer of antiquities is more regulated, and closer to how I imagine modern archeology.
Was there a shift in methods? When? What brought on the change?
Perhaps these other topics will answer your inquiry
https://www.reddit.com/r/AskHistorians/comments/2wzce9/exactly_what_time_does_the_crossover_occur/
https://www.reddit.com/r/AskHistorians/comments/1iyq5h/how_much_of_archaeology_brings_up_treasures/
hi.. there are a few archaeologists here, but this may be worth x-posting to /r/AskAnthropology or /r/Archaeology