How does the direction of the electric current, moving across a battery powered device, differ from the direction of travel in the ac current sent to homes and businesses? .

A battery is a constant source of potential difference which will produce constant potential difference across the resistance.

Now since the polarity of battery will always remains the same and the magnitude of potential difference also remains the same so here we can say that the battery will produce constant current across a resistance.

So here we will say that the direction of current in any resistor will remain the same and will not charge when it is connected across a battery

now when resistor is connected across a AC source the the AC source always changes the polarity of the battery periodically as well as the magnitude of EMF is also not constant.

Since the polarity is changing here in AC source with time so we can say that the direction of current in AC source will change periodically and hence current in the resistance is not constant in direction as well as in magnitude.