Medical devices and related methods are disclosed for delivering an implant into the body of a patient. The devices include a damping assembly having an internal chamber with a varying inner diameter, and a piston slideably disposed therein. The piston is operably connected with a catheter portion such that movement of the piston is associated with concurrent movement of the catheter for implant delivery. The varying inner diameter of the damping assembly results in a decrease in damping force during implant delivery, thus providing a substantially constant velocity of implant delivery.