I took a composite style of karate in the late 70s / early 80s, and we were taught jab and cross but we also cross trained in boxing . . . it was hardly traditional karate.

Recently my kids did TKD. They were also taught jab and cross. But TKD is usually not taught the way is used to, for example full contact sparring was the standard in my kid's school.

I don't recall jab or cross being used in any karate/TKD forms I've seen. Are they part of traditional karate/TKD or are they additions from boxing?

I'm with you, I dont recall jab/cross (western boxing style) in traditional TKD (took it as a teenager).
That said, there are so many belnded systems out there now, as most martial artists want to be cross trained. Some Karate classes are even teaching jiu-jitsu/grappling to some extent as a part of the cirriculum.
For example, my "kickboxing" class is a mix of kempo karate and muay thai.

My teacher was training in traditional jujutsu and kickboxing by different teachers but pretty much at the same time, so when I learned from him we did jab/cross more than shoto or ridge hand strikes. It blended nicely together.

What matters is what you want; do you want an exclusive historical style based on ways people no longer attack or are you happy with a modern style that takes in the way modern assults happen? Neither is wrong, it is just what you want out of the martial arts.

I am asking because I'm trying to get a better understanding of the traditional makeup of karate (and other arts). The history and evolution interest me. Now I train BJJ only, if I had more time, money and energy I'd train in other stuff as well.

I'm pretty sure we trained jab-cross when I did karate (red belt in Chuck Norris' UFAF in early 1990s), but I've studied a lot of other styles since then, so can't say for sure, and I don't remember any of the UFAF forms. We were also learning a little bit of BJJ back then, so it wasn't exactly “pure” karate, not that it ever really was (Chuck Norris' version of Tang Soo Do, which itself is an amalgam of Korean and Japanese styles).

Punching to the head was illegal in a lot of tournament styles back when I was studying karate, I'd imagine they don't practice jab-cross combos much.  I have no idea what the rules are today. I gave up on karate when I discovered kickboxing, really no comparison when it comes to effectiveness.

Hey, new member and just found this thread:

FWIW, in my style we learn both the "do" and "jitsu" of the techniques... Do = the way of, and Jitsu = practical application of.

That being said we train all kinds of combos with the jab/cross being a regular player. Jab/cross/hook/low kick is a favorite of mine Our style also incorporates a bit of jiu-jitsu (Japanese) so about 3-4 years ago we added a full jiu-jitsu class. We don't train for point fighting so much as we focus on a working system for self defense and to that end traditional karate and jiu-jitsu dovetail beautifully.

In the late 1800s Okinawas most famous Karate-ka wrote in a letter to a friend in which he lamented that Karate was now becoming a sport rather than fighting.  With every generation of teachers since then it continued on that path.  Remember than when Funakoshi introduced it into mainland Japan he revised it to be taught for P.E. at junior high school (not a joke).  Then there's also the situation of the 'real' version which is taught only to family and few selected others and the public version which is taught in classes.