A good question which I hope sparks up a lot of interesting discussion.
I have to make the caveat that I have started barbell training only recently, about six months ago. So my experience is limited and my insights so as well. Still, I am about to hit decent weights with the barbells, as in planning a five wheel squat for next week and having deadlifted more already, so I think there's some merit to my opinion.
Before the barbell I did mostly kettlebell training: swings, cleans, presses and front squats mostly. I have about five to ten years of experience with the kettlebell, though my training has been leisurely instead of goal-oriented apart from the pressing.
I think the kettlebell training has had an influence on how my barbell training has developed. The years of training with the kettlebells have taught me the core principles, like tension, bracing, breathing etc. I developed a good hinge. My body awareness was high. So, I wasn't surprised I managed to progress adequately with the barbell. But I don't think the kettlebells I had, at max a 40kg or 2*24kg, prepared me at all for the sheer absolute weight I was about to get involved with, apart for the overhead press. So there was little carryover that way.
But, I have found the carryover does work the other way. I have done little kettlebell training due to circumstances lately, but have tested them out some times. The first time I tried swinging my 40kg again I had deadlifted 500 pounds. The old bastard was
light. I think I made more progress in the swings with two months of barbell training than I would have made in two years of kettlebell training. If not more.
In my case, the barbell benefits don't stop at making things
light. I had done plenty of bodyweight and kettlebell front squats. They had really helped me move better and enabled me to easily do crouch walking at work for length, for example. But, I still had some problems in the hole. I needed a counterweight to be comfortable with my heels on the ground. I originally thought it was an ankle mobility issue, but I flew perfectly through the tests. So I took it to be case of acute bellyitis, as in being too fat for the thing. Not so anymore after the barbell back squats. My waist circumference has remained constant and my weight has come up a bit. I can squat down easily without a counterweight for minutes now instead of seconds. My shoulders haven't been better in a long time since doing a lot of barbell pressing, rowing and back squats. So, by no means do I agree with the viewpoint that barbell training makes one immobile or inflexible. I think it's the opposite. And don't even try to get me started on back pain...
If it doesn't strike out as obvious yet, I have, due to my personal experience, become a true believer in the barbells. I think they're the best thing I've ever done, training wise. The kettlebells have carryover to the barbell world, but I think it is greater the other way round. So I agree with
@Bro Mo . I have yet to make a new cycle of pure kettlebell training and then come back to barbell training and see how the carryover goes. I don't know when that will happen. The thing is, I don't think I ever want to do it.
In the end, I'll just leave this article link here:
Absolute Strength Is the True Master Quality