If an organism that reproduces asexually clones itself through this process and makes genetically identical organisms, then how can it be called "descent with modification" or be related to evolution (where change in genetic code of the successive generations is the key principle)? Does it mean that asexually reproducing species do not take part in evolution?

genetic variation/mutation still occurs in animals that reproduce asexually which is why they have evolved into other species over many millions of years. evolution acts equally on all living species and for the same reasons (random genetic variation followed by selection of altered phenotypic traits).