--- Ulysses S. Grant
I had him! I swear, I had him in the palm of my hand. And then he slipped away.
--- Me, between bouts of banning my head on the keyboard
So how did the CSA Army of Northern Virgina, trapped in Carteret NC (Morehead City), a force of 20,000 men, slip between the fingers of my Army of the James lead by Meade with 72,000 on duty on a cold and wet spring day in 1864? With Meade and two corps in Onslow NC and the only places to which the ANV could retreat were either into the forts allong the Hatteras or into the swamps of Craven NC, who's entry with cavalry alone cost 16 days, let alone to consider that the Pamlico Bay lays between them. There's no way in hell that they can get away before I'm on them and seeing to their demise. Or so I thought.
After finding myself in a similar situation on the Mississippi, not trying to escape, but once again on the advance, trying to get at the enemy hiding the the Yazoo swamps, I finally figured it out. And since this was such a revelation for me I thought I'd share it for the benefit of the rest of us n00bs - eh.. fresh West Point lieutenants.
Here's the situation. I'm on the Hayne's Bluffs overlooking the Yazoo into the Tallalula Bayou where the enemy is. But it would take 16 days just to enter Tallalula Bayou even if the Yazoo River weren't in between.
With the addition of crossing the Yazoo I'm already up to 19 day. That takes far too long.
How do I get Slocum's corp over the Yazoo before Johhny Reb slips into the murk of the bayou and away? I could run to the river transports awaiting me in Vicksburg, load up, sail out and debark right on top of them. I'd get there in the best condition, but that would take three whole turns(!!). How about force-marching. There's a 75% chance of success of that, but it still takes 15 days. Still far too long.
And that's when I had the epiphany. If I didn't have to go to Vicksburg to load onto the transports, if they were already right there, and if I could got off them in the same turn that I got on, I wouldn't even need to sail up or down the river, just get across; if. But I can! Just click the 'use river transportation' button and off you go. It's that simple and I'll be on Johnny Reb's tail in six days.
And if that's still not fast enough for your taste, you can still force-march still with a 75% chance of success and be on him in 5 days, though a bit winded.
In certain situations using the river transports can border on black magic.
It only costs 1 day to get on them and five to get off, always. It doesn't matter where you get off, it still just costs 5 days. So if that swam region cost 22 day to walk into in the the middle of winter plus a few to cross the river in between, it will still only cost you 1 day to get on the river transports and 5 to get off. And on top of that, if the terrain on the other side of that God forbidden swamp is good marching ground and only costs for example 6 to enter, you can be over the river, through the swamp and onto good solid ground in 11 days! and still have time to keep moving. 
Now that's a bargain for the price of using those precious generic river transports. And don't forget. They are always there, even deep in enemy territory.
-- Captain Orso
Edit: Exception to the 5 day unloading is if the unit(s) unloading is/are inactive ie they've got a case of the brownies
ouet: Maybe that'll wake them up 
My experience anyway...unless thigns have changed recently. 
But it usually takes so long for me to get my attack coordinated that the enemy is starting to starve by the time we get there. It's like besieging an entire swamp. I don't think Athena quite knows what to make of me. 