since a 30 year bond discounts 30 years of cashflows and those cashflows directly incorporate this compounding yield - its price moves more with yields than a comparable, shorter time to maturity bond
regarding 1 year vs 30 year bond - imagine yields rise by 2%: β the price of the bond maturing in 1 year declines by discounting those 2% from 1 year of cashflows β a 30 year bond discounts for 30 years of cashflows