Over 380 years ago this festival originated from celebrating the move of the shrine and walking through town carrying a drum on a wooden cart.
On the 7th it continued to children who walked in the guise of people from olden times and warriors on horse back, and 19 parade floats were pulled along.
When I started to ask this question I had no idea how to understand に続いて. My current guess is that に続いて pairs with 始まりました from the previous sentence, i.e. the festival originated (始まる) from doing X but continues (続く) to this day by doing Y.
Have I understood this correctly? If I'm right it feels a bit strange that a specific date is given rather the just 今 or この頃 or something like that. Obviously I can understand that they want to tell us what date the festival was, but my English translation seems awkward. I wonder how natural the Japanese is?