We present a high-order cut finite element method for solving coupled bulk-surface convection-diffusion equations in evolving domains, modeling for example the evolution of surfactant concentration in a two-phase fluid [1]. By applying Reynolds' transport theorem to derive the weak formulation, our method naturally conserves the global mass over time [2, 3]. The method employs space-time discretization, but the integrals in time are approximated by quadrature rules to produce schemes resembling time-stepping methods and avoid constructing the full space-time domain [4]. Quadrature rules that result in optimal order of convergence are used [5]. Furthermore, a more efficient stabilization procedure for cut finite element methods is presented by partitioning the time-dependent domain into macroelements. In this approach, stabilization is applied only where needed, leading to an increased sparsity of the resulting system matrix while maintaining control of the condition number and the error independent of the position of the evolving domain relative to the mesh. The key result is an unfitted finite element method that combines mass conservation with high-order rates of convergence for convection-diffusion equations in evolving domains.