substitute, change, convert, replace, period, age, counter for decades of ages, eras, etc., generation, charge, rate, fee
代 centers on one thing standing in for another: substituting, replacing, or taking a turn. This extends naturally to a span of time (an era or generation) and to a fee or charge as something given in exchange.
period; epoch; era; age
representation; representative; delegate; delegation
substitute; replacement; substituting; replacing
present age; present day; today
generation
age; era; period; date
representation; agency; proxy
ancient times; distant past; antiquity
change; alternation; rotation; replacement; substitution; shift; relief; taking turns
first generation; founder
acting as agent; acting on (someone's) behalf; executing business for
present day; modern times; recent times
substitution
successive generations; successive emperors
metabolism
price; cost; charge; payment; bill; fee
proxy; agent; substitute; deputy; alternate; representative; attorney
compensation; reparation; indemnification
substitution
pinch-hitting; pinch hitter
for generations; from generation to generation; generation after generation; through the ages
pronoun
previous generation (of a family); previous head of the family; one's (late) father
thousand years; thousand generations; thousand ages; eternity
代 combines 亻 (person) with 弋, which likely contributed the sound. The idea of a person standing in for another gives the core meaning of substitution.
A person 亻 holding a stake 弋 marks a spot where someone else will take over. Picture a relay runner passing a stake to the next runner: one person replaces another.
For ダイ, imagine a substitute runner named Dai who takes the stake and runs the next leg: Dai -> ダイ.