Question: I have seen some general comments online about running a 6V DC motor on 9V Battery being OK. 
I am looking to run an air mattress pump originally designed for 4x 1.5V D batteries (6V DC) from a universal car laptop adapter. The lowest output voltage on the car adapter is 9V.

Nothing on the air pump motor mentions input power specs specifically. The only markings I can see are the following 3 lines and I get no hits googling each string.
hb-138-101 
rs-380ph-6016rb 
630301

The alternative would be to use an AC inverter and an DC 6v 2A adapter. It sounds like overkill to go from 12v to 120v AC just to get 6VDC, but so is running a car engine. 
Will I burn the motor if I ran it on 9V DC?

Answer: 
Yes, the same number of electrons is being transferred per unit of time in each case.

Nonetheless, more power is being delivered to the load in the case with higher voltage.
Practically, I have two situations where that amperage is the same. My Electric Vehicle charges at 16 Amps with either a 110 Volt charger or a 220 Volt charger. The 110 Volt charger takes twice the time to charge the car battery at the same amperage. When I look at my electric company's energy use in kWh the total cost to charge my car battery is approximately the same. Hence with 1/2 of the voltage, only 1/2 of the power is delivered per hour.

Back to the water analogy. Is it the case that with the 220 Volt charger, my "water flow" is the same per hour, but with the 220 Volt charger I am using "two 110 Volt hoses" (hence 2x the amount of power per hour compared to a 110 Volt charger)??