I guess:
You are talking about Alternation of Generations - gametophyte and sporophyte. And the production of gametes is NOT asexual. So the answer is: yes. Meiosis always "scrambles" DNA in chromosome pairs when crossing over occurs. It is sometimes called recombination. In evolutionary terms it is known as 'meiotic drive'.
Somewhere you got the word 'spore' misplaced in your understanding. Flowering plants produce meiotically generated spores (gametes): pollen grains, for example. Not part of asexual reproduction.
Cell division or mitosis is the driving force for asexual reproduction. Gametophytes like molds can make spores by mitosis, even though they are haploid.
All of this gets messy and confusing to students when you look at fungi, because most times, the organisms we can see are gametophytes - they have half the normal number of chromosomes.