update every row with a different random datetime of a range

The question asked here is

Update every row with a random datetime between two dates

to update with the same datetime.

What I need is to update every row with a different random datetime within a range.

With:

strftime('%s', enddate) - strftime('%s', startdate)

you can get the difference in seconds between 2 datetimes.
With:

abs(random() % (strftime('%s', enddate) - strftime('%s', startdate) + 1))

you can get a random integer, greater or equal to 0, that is less than or equal to the difference between the 2 datetimes in seconds.
What you can do is add this random number of seconds to the starting date of your range to create a random datetime within that range:

update tablename 
set datecol = datetime(
  startdate, 
  abs(random() % (strftime('%s', enddate) - strftime('%s', startdate) + 1)) || ' second'
)

datecol is the date column in your table
startdate and enddate are the boundaries of your range.