The best way to flush cooler lines and coolers is with a cooler flush machine. These machines are designed to run hot ATF through the cooler, switch directions quickly to loosen any debris, and carry it to a filter. The better machines have screens to detect how much debris has been removed.
Another feature is the pressure. If it’s too high or restricted, then the machine will shut off. We strongly recommend the hot flush machine for all shops. Ultimately, it is a come-back saver!