DLA and ESA are 2 seperate benefits . DLA is not means tested , you can be a millionaire and still get it.
Some ESA is means tested , you can get ESA but not qualify for DLA, the level of disability needed to get DLA is a lot greater than the level for ESA .
The person with £120,000 you mention would not get 'income based' ESA but if they get DLA it will not be effected.
To confuse things further there are 2 types of ESA , contribution based and income based . Anyone who has paid full NI can get contribution based ESA for a year if they meet the medical reqrements , that one is not means tested.
But after a year you can only get income based ESA which is means tested.
But DLA is a seperate benefit it does not change to ESA and as I said is not means tested.
Where you are getting confused is that if you no longer qualify for ESA you go back to JSA ( Jobseekers allowance) which is less and requires you to 'sign on' every 2 weeks.