Here is the reality check. City Class got boats had a top speed of 5 knots. The current of the Mississippi River is (+/-) 3 knots.
To cruise up river a City Class gunboat burned a ton of coal / hour. A barge of coal was lashed onto one or both sides. The entire crew, including officers, were engaged with n transferring 100 pound bags / baskets CV of coal.
A tugboat would be towing from the bow. With the stokers, water tenders & black gang tending the three boilers the gunboat's wheels thrashing up a froth, two maybe three knots could be made relative to the river bank.
By way of perspective, a small girl peddling full tilt can make two knots.
As at Vicksburg, returning back up river under fire of the batteries was not possible.
Another aspect is that the bounty of Mississippi Valley flows downstream to New Orleans. Blessings flowed downstream from Cincinnati. Pemberton's supplies came by rail from Georgia. Months before Grant crossed the river Pemberton reported that he was cut off from supplies from the Trans-Mississippi,
You could argue that the Westrrn war was powered by gravity.