Grasping and manipulation are complex and demanding tasks, especially when executed in dynamic and unstructured environments. Typically, such tasks are executed by rigid articulated end-effectors, with a plethora of actuators that need sophisticated sensing and complex control laws to execute them efficiently. Soft robotics offers an alternative that allows for simplified execution of these demanding tasks, enabling the creation of robust, efficient, lightweight, and affordable solutions that are easy to control and operate. In this work, we introduce a new class of soft, kirigami-based robotic grippers, we study their post-contact behavior, and we investigate different cut patterns for their development. We follow an experimental approach in which several designs are proposed and employed in a series of grasping and force exertion tests to compare their capabilities and post-contact behavior. The results of such experiments indicate a clear relationship between degree of reconfiguration and grasping force, and provide key insights into the effect of the cut patterns in the performance of the designs. These findings are then used in the design process of an improved version of multi-layer, disposable kirigami grippers that are fabricated employing simple 3D printed layers and silicone rubber using the concept of Hybrid Deposition Manufacturing (HDM). A series of experimental results demonstrate that the proposed design and manufacturing methods can enable the creation of soft, kirigami-based grippers with superior grasping capabilities that can handle delicate, contaminated, and everyday life objects and can even be disposed off in an automated way (e.g., after handling hazardous materials, such as medical waste).