We investigated properties of the Riesz transform and large time behaviour of derivatives of the heat kernels corresponding to left-invariant operators on Lie groups. We proved that higher order Riesz transforms for such operators are bounded if and only if the corresponding Lie group is a direct product of a nilpotent group and a compact group. We also obtained results concerning connections between boundedness of the Riesz transform and the behaviour of derivatives of the heat kernels. The finite speed propagation techniques are important part of the proof.