If transportation was much slower in the past, then how did big empires arise?

The best empires had some pretty ingenious ways to communicate/ move troops and trade.

The Romans became great road builders because over broken ground a legion could only march maybe 2 mph/ about 15mi / day but on a road a legion could march up to 30mi/ day.

The Persian empire actually had a royal message system and postal service (the US 'pony express' was a much later copy). The Persians were also big road builders, so a messenger with mail on horse on good road could go as much as much as 100 mi/ day & there were rely horses/ rely riders/ to take over as soon as horse or many got tired. They could even go through the night, so 200mi/ 24h day.

The British did everything by ship, all they needed was wind and they went 24/7 with no effort/ 300mi per day.

Along with this they appointed governors and troops, stationed them around and changed them regularly so they'd stay in line.

P.S. The Mongols lived (literally) on their horses/ on horseback. Each warrior was required to have 3 horses, so when 1 tired/ got hurt he'd just switch, so a Mongol host could cover 200mi a 24h day across plains.     

Like the British empire, they had Governors to run the Countries we Occupied they were the Bosses of the Country who answered to the Reighning Monarch