How does carryover work?
If I get a stronger chest with any random exercise does the bench press go up?
Or is there the need to do similar exercises like push ups and dips in order for carryover to happen?
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Strength is neural but having muscle/size helps. If you're relatively heavy (the heavier you are the more weight you're working with which will definitely carryover to more weight on the bench press assuming you're other variables like nutrition/form are in check) and train with pushups/dips for many reps per set with progressive overload, you're bench will definitely go up. Many athletes that have never benched before can come in and throw up a lot of weight because of their training regimens carrying over (wrestlers, gymnasts etc.) How much weight you throw up and what carries over depends on a lot of things from mass, athleticism, form etc. So if you're new to the bench and can't put up 135, doing push ups and dips will definitely make your bench go up.
However, just remember that eventually you hit numbers on the bench press where specialized training comes into play. For example, a person starting to hit three hundred pounds or more on the bench isn't going to get a lot out of push ups in a time effective manner. He'll have to design a training regimen to hit those higher weights targeting variables like bar path, endurance, weights, stabilizers, frequency, reps, accessories etc.