Then can I assume that in Asexual species, the 'offspring' are all the same??
Asexual species evolve slower than granite, which is why the invention of sex about 800 million years ago sparked a notable speedup of evolution. I think we talked before about the Vendian "sizzle" that preceded the Cambrian "explosion."
Mutations still happen and lines of descent still branch in asexuals but there's no recombination. Lines never cross. Organism A never exchanges genetic material with organism B even to reproduce. A just clones or buds another little A, etc. If it takes 10 mutations to get a feature, all ten mutations must happen in the same line of descent with nothign ever coming in from outside.
Some exceptions exist for bacteria, which rather freely exchange genes laterally even between highly unrelated species. Another exception is that viruses can infect multiple species of higher life, carrying their viral DNA into the genomes. The implications for evolution are still being argued over.