as far as I know, there is no birth and death of elements implemented in Comsol yet. But there are a lot of nice papers using the material properties to get the same result. In this case, you simply mesh your whole part and define the properties time dependent. Hence, in the region where the material is added, you have the normal material properties and in the region where the material is not added yet, you have all material properties set to zero (this simply means that no heat will propagate within this elements). I hope this helps :).
Hi, In my opinion, you can use the ALE. The information about ALE can be found at the reference manual inside the chapter deformed geometry and moving mesh.