Those who bring up this claim are actually mixing up sex and gender. See Objection: Sex Cannot Change.
Gender obviously changes all the time! When you wake up groggy with smelly breath, tangled hair, and in your pajamas you have one gender. When you shower, put on decent clothes, and make yourself presentable to the world, you have another gender. When you go out on a date in formal clothes, wooing your date, you have yet another gender. That's why we have different gendered words - man versus gentleman and woman versus lady.
Even if you disagree with my terminology, every culture recognizes at least one change in gender which everyone undergoes: From child to adult during puberty. A girl becomes a woman and a boy becomes a man. Even if you insist on equating sex with gender, a person's sex changes too as genitals enlarge, sexual response cycle is enabled, secondary sex characteristics appear for the first time.
In a sense, everyone changes their gender. In another sense, gender never changes. Gender identity (one's innermost sense of their gender) never changes. A person's gender identity in childhood is the same as on their deathbed.