At the end of the day, you have to be paid for all the time you actually worked. If, when you get your paycheck, it comes to minimum wage times the number of hours you worked, and you received any overtime that might have been due, Federal law will consider that you have been correctly paid. The TWC *might*, if you get the right claims adjuster, take issue with that working out to a lower rate of pay, but the last report I had was that Texas was less interested in wage claims than they were at one time. Texas has no laws regarding the reduction of wages, the timing thereof or any notice related to such a reduction. I'm not saying that a wage claim over such a reduction would absolutely fail but it's by no means a guaranteed win either, not in Texas.
What is more to the point is that, with limited exceptions to do with serious medical conditions, you can be fired for having too many "family emergencies" that require you to leave early. Unless you qualify for FMLA AND one or more of your kids has a serious health condition as defined by the FMLA statute, AND you have submitted medical documentation to that effect AND you have been approved for FMLA leave on an intermittent basis AND the reason you are leaving early has to do with this medical condition, family emergencies are not protected under the law and you can be disciplined for it up to AND INCLUDING termination. They can't simply not pay you, although as indicated above it's possible to play games with the rate of pay. But you can definitely be disciplined in other ways.
So if this is happening often, I recommend in the strongest possible terms that you get some kind of child care backup because leaving early, with or without permission, too often can put your entire job in jeopardy. And your employer is the one who gets to decide how often is too often.