When I swapped over to synthetic in my Subaru, the car was at 120k miles and it was a little process. Swapped in the synthetic, drove 500 miles and changed the oil and filter. It was BLACK. Ran another 1,000 miles, changed the oil, it was BLACK again. Ran another 1,000 miles and changed, changed the oil and it was mildly dark. Ran 3,000 miles and changed it and the oil literally looked the same as when it went in. Sent sample to Blackstone. They advised 7,500 miles. Ran 7,500 miles and changed oil/pulled sample. The additives were almost the same as a virgin sample, recommended going to 10,000 miles.
Oil always came out looking almost the same as when it was poured into the motor from then on.
Now, I did the transition in this manner because I did not know the health and status/maintenance history of the engine.
Could i have just done 3k interval? Absolutely. My goal was to get the oil clear which means the internals were clear of any gunk and deposits, then take a used oil sample to verify the health of the internals.
Does the transition have to be done this way? No. 3-5k intervals are fine.