I think low EQ may be be partly learned and partly innate or caused by early parenting. I don't know the specifics of your case, but with my older sister, it really bothered me when we were little that she had no empathy for other kids, especally ones younger than her. Once a boy younger than her fell off his bike and started crying. My sister pointed her finger and said. "Ha ha ha!" to mock him for crying. I don't know if it qualifies as low EQ because she knew enough to not do that when adults were watching. I only suspected low EQ on our 30s when her temper tantrums increased to about 15 times a day, screaming over stubbing a toe, having to make her own food or coffee, and other every day things. I see low EQ as having the self control and self awareness of a toddler. People with low EQ also seem to project their feelings on others rather than recognize they are the angry/insecure/envius one.

The thing that is alarming is that you have young kids. I know that my stepfather's low EQ definitely made my sister worse. Once he was in our lives, my sister went from child bully to someone really violent and cruel. This is the sort of thing kids mimic from adults because they view parent behavior as an example of what is normal. I know that as a kid, I copied my sister's narcissistic remarks about other people because I mistook what she said for humor. The impact on your kids would probably be a good thing to bring up with a therapist.