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When I started pumping a new Pumpico floor pump, the shaft slid down doing nothing for about 12 pumps and then kicked in for 2 hard pumps before it did nothing again for another 10 pumps. It's a schrader valve. My tires needed only a little air. Do I have a defective pump?
When I pump the shaft when it's not connected I feel no air coming out of the tube. Is that normal?
That sounds like a broken seal in the pump.
I had to google the brand as I'd never heard of it before. There are a few amazon reviews that also state the pump didn't work on arrival. Link to Amazon
Given that evidence, and the tragedy that is the Pumpico website: https://thepumpico.com/ I'd suggest returning the pump and buying something from an established manufacturer of cycling products/accessories.
