Yes you can put containers on a flat top deck barge and tow it between locations by wire boats.
In fact it is done every day in the US, between JAX and PR, from USWC to Hawaii and Alaska.
Why it is not done as a feeder service between ports along the coasts? I don’t know, but apparently it has been tried
To put containers on existing river barges is a different proposition. To be efficient they need to have box shaped holds and “open hatch” construction, The hold dimensions has to be suitable for a certain number of standard 20/40 ft., both in length and breadth. Load capacity of the tank top has to be sufficient for the intended max. weight/point load that may be asserted by fully loaded containers.
The height of the container stacks MAY be limited by bridges along their intended route.
If the barges are to be pushed by Pusher tugs the wheelhouse height may also limit the stack height.
(Since the existing US pusher tugs have a “fixed” wheelhouse height, line of sight MAY be the limiting factor)
The normal way is to use STS cranes and stack the containers on the terminal, then use same STS cranes to load the feeders/barges, or to move the containers to a different wharf and load with smaller cranes.
Mobile wharf cranes are used in many places:
In an ideal world they would be filled with goods and returned to a main line terminal for export.
Unfortunately it isn’t an ideal world, so sometime empty containers has to be transported half way around the world to be repositioned where they are needed.
Who take the expenses for this transport?
It depends on who owns the containers. Some are owned by shipping companies, others by leasing companies and some may even be owned by large shippers or importers.
In any case there are some form of agreement, or contract, stating how this is arranged. In most cases the consignee has a fixed number of days to collect, transport, un-stuff and redeliver the container to an agreed depot or terminal. Failing to do so and a demurrage charge is applied.