  A mass-conservative high-order unfitted finite element method for
convection-diffusion equations in evolving domains is proposed. The space-time
method presented in [P. Hansbo, M. G. Larson, S. Zahedi, Comput. Methods Appl.
Mech. Engrg. 307 (2016)] is extended to naturally achieve mass conservation by
utilizing Reynold's transport theorem. Furthermore, by partitioning the
time-dependent domain into macroelements, a more efficient stabilization
procedure for the cut finite element method in time-dependent domains is
presented. Numerical experiments illustrate that the method fulfills mass
conservation, attains high-order convergence, and the condition number of the
resulting system matrix is controlled while sparsity is increased. Problems in
bulk domains as well as coupled bulk-surface problems are considered.
