The Mongols established defined routes called a Yam much like the Pony Express routes of 19th century western America.
Relay stations were used to give food, shelter and spare horses for
Mongol army messengers. Genghis Khan gave special attention to Yam
because Mongol armies traveled very fast, so their messengers had to
be even faster, covering 200–300 km per day.
(between 100 and 200 miles).
Another entry gives an overall time:
When the great khan died in Karakorum, news reached the Mongol forces
under Batu Khan in Central Europe within 4–6 weeks thanks to the Yam.
So 4-6 weeks to cross the earth based empire. You might assume similar time to cross your star empire, then divide by that distance to get your FTLs rate of travel to produce similar delays.
An example extrapolation (I'll make some assumptions):
Our Galactic Mongol Empire controls this arm of the galaxy, with Earth as its capital, and we need to send a message to a general near the galactic core, and this will take the same time (4-6 weeks) as the above mentioned message.
- Earth to core distance = 8000 Kpc = 26,092 LY (round to 26100)
- Time for Mongol message (pick 5 weeks) = 5 * 604800(seconds per week) = 3024000
gives us 26100/3024000 = 0.0086309523809524 LY/sec
multiply by 60 gives us 0.5178571428571429 LY/minute
The last value we can round(if desired) to 1/2 LY per minute or 2 minutes / LY, so the OPs example of 3-10 LY separation means travel between typical worlds will take 6 to 20 minutes, to simulate an expansion of Mongol rates of travel to a galactic empire stretching from earth to the galactic core.
Note your mileage may vary.