Water softeners utilize water pressure to perform each regeneration procedure. Each regeneration procedure excluding Brine Fill, will result in water flow to drain. For drain water to flow some distance upward to drain, does not require a pump, but the vertical distance will be largely determined by incoming water pressure.
To ensure the required flow rate to drain, when the water treatment device's drain line distance exceeds 15' horizontally, or when routed above the softener, the usual recommendation will be to increase the drain line diameter (ie: use 3/4" for a softener equipped with a 1/2" drain connection).
Because a water softener processes potable water, the softener's drain line is not to be connected directly to the home's drainage system, but is to be discharged through an air gap device to ensure no contaminated discharge can contact the softener drain line and backflow into the softener if there ever is a sewer backup.
Image below shows a softener's drain line connected to an airgap device installed above the P-trap.