I do the same "bottle trick" with a syringe and it seems to clean out the cart nicely

Did you watch SnowDragon's video on filling carts? We have the most awesome group of techies here so your every question will be answered
Also, I first blow from the battery end, after cleaning off the threads and cart while holding a tissue to the mouth end. Then I sometimes have to reverse the process and blow into it via the mouth piece
Word of advice...ALWAYS make certain you have cleaned the cart of juice that might have dripped, spilled, leaked over...etc. You do NOT want to wrap your lips around ejuice
