Partial differential equations posed on moving domains arise in many applications such as air turbine modeling, flow past airplane wings, etc. The time-dependent nature of the flow domain poses an additional challenge when devising numerical methods for discretizing such problems. One alternative when dealing with time-dependent domains is to pose the problem on a space-time domain and apply, for example, a finite element method in both space and time. These space-time methods can easily handle the time-dependent nature of the domain. In this talk, we present a space-time hybridizable discontinuous Galerkin method for discretizing the incompressible Navier-Stokes equations on moving domains. This discretization is pointwise mass conserving and pressure robust, even on time-dependent domains. Moreover, high order can be achieved both in space and time. For fluid-rigid body interactions, we have developed an edge flipping meshing technique that can connect the spatial meshes with slightly different topologies to create the space-time mesh. The rotation and/or the translation of the rigid body is described by ordinary differential equations coupled to the fluid equation through the lifting and the pitching force. Out edge flipping algorithm allows us to use precomputed meshes to incorporate arbitrary rotation. Numerical experiments will demonstrate the capabilities of the method.