claiming =>claim 一般现在时,强调的是后面做的内容
然后加个that读起来会舒服点
people claim that they are doing ancestry testing.
或者
claiming =>hava been claiming 现在完成进行时,强调claim这个动作
people hava been claiming that they are doing ancestry testing.claiming 算是动名词吗people claimingthey are doing ancestry testing.

这个句子不对的，分不出主谓宾
语法上对也要这样people are claimingthey are doing ancestry testing.
而即使这样，claiming也不是动名词。
动名词是动词名词化，比如：Eating apples makes us healthy.