What do L2 scaling solutions mean for mining?

My naive understanding of L2 scaling solutions is that transactions are "batched up" off chain and then submitted to L1 for verification.

If that is the case, then what does this mean for mining? More specifically:

• Will there be fewer (but larger) transactions to process on L1 as users move to L2?
• Will mining profitability decrease if there are fewer transactions to process or will the fees increase proportionately to the size of the transactions (therefore compensating for the lack of L1 transactions)?

I'm sure not all L2 solutions work this way (and my understanding is likely wrong even for those that do some kind of batching - if so, please correct me).

N.B.: I'm aware Eth 2.0 will render all of the above moot. However, I'm interested in the effect L2 will have in the short term.