I am unable to find much work regarding what happens after mining ends.
It depends on what you mean whan you say "mining". The process of adding transactions to blocks, solving for a hash of appropriate difficulty, and claiming the block reward will never end so long as bitcoin continues to exist.
What will eventually end (somewhere around the year 2140), is the block subsidy. The block reward consists of the sum of the block subsidy and the total of all transaction fees voluntarily paid by all transactions that the miner chooses to include in the block that they solve.
Over time, the block subsidy (currently 25 BTC) is reduced. It is cut in half every 4 years. This means that if bitcoin continues to increase in usage in the future, the portion of the block reward that comes from the transaction fees will likely increase while the portion from the subsidy will decrease. Eventually more of the reward will come from fees than subsidy (though a significant portion will still come from subsidy). The percentage that is derived from subsidy will taper off until eventually the entire reward is coming from the fees.
Everyone will flock to the largest transaction processor since they will likely offer the lowest rates. As their market share grows they will bankrupt all the other processors by offering even lower rates. When the smoke clears they will look around for any other competition like VISA, if they still exist. If VISA has a rate of 2%, they will offer 1.9%. They will be so large that the cost of entry for competition will be prohibitive.
When you create a transaction, you don't choose the miner (or mining pool) that will confirm your transaction. You simply broadcast it to all your connected peers. Miners choose which transactions they want to confirm. Any transaction that is not confirmed by the "largest transaction processor" ( I assume you mean "largest mining pool"?) will be available to be confirmed by smaller mining pools. While the smaller mining pools may not solve as many blocks, they won't have to split the revenue among as many mining units and therefore will be able to pay a larger portion of the block reward to each participant.
The solution: In processing transactions there must be an incentive to maintain an optimally sized network for competition and security of the blockchain. It must be built-in.
It is. It's called the block reward (coupled with an automatically adjusting "difficulty").