Electromagnetic actuators provide fast speeds, large forces, high strokes, and wide bandwidths. Most designs, however, are constructed from rigid components, making these benefits inaccessible for many soft robotics applications. In this work, we develop a new soft electromagnetic linear actuator using liquid gallium-indium for the conductor and neodymium-iron-boron and polymer composites for the permanent magnet. When combined in a solenoid configuration, high strokes can be generated using entirely soft components. To emulate the pulsing motion of Xenia coral arms, we develop an additional soft flexure system that converts the linear translation to rotary motion. The design and fabrication of the electromagnetic actuator and compliant flexure are first described. Models for the magnetic forces and the joint kinematics are then developed and compared with experimental results. Finally, the robot dynamics are analyzed using stochastic system identification techniques. Results show that the compliant actuator is able to achieve an 18 mm stroke, allowing the soft arms to bend up to 120 degrees. This further enables the tips of the arms to traverse an arc length of 42 mm. Bandwidths up to 30 Hz were also observed. While this paper focuses on emulating a biological system, this highly deformable actuator design can also be utilized for fully-soft grasping or wearables applications.