The argument goes like this:
A business employs some people at $10 an hour (which is the minimum wage in this scenario) and other workers at $12 an hour. These are the two lowest pay levels at said business.
Now, the minimum wage gets raised from $10/hour to $12/hour; what happens to the employees who were paid $12/hour before the minimum wage is increased? Their wages won't be increased (or so the argument goes) any time soon, and wage increases for them will actually be delayed, yet they have seniority and experience over the minimum wage employees.
Is this a valid argument?
No, it's not.
That person receiving $12/hour will, when the minimum wage gets boosted, immediately become underpaid. This means the employer will face increased pressured to maintain the company's labor structure. The employer can continue to pay the minimum, but as the market adjusts, the employee will be sorely tempted to leave their current job in favor of one that, once again, pays higher than the minimum.
I think that depends on the local labour market. A call centre worker might be low skill, but still have something minimum wage earners don't have (well spoken, willing to put up with abuse etc.)
As long as there are alternative, higher paying positions available for those modestly skilled workers, I agree, but I'm not so sure that is the case in economically depressed areas.