Through changing the federal fund rate(short-term), does this empirically impact the longer end of the curve? If so, what is the logic or transmission mechanism that caused the change?
Yes, it does. This topic is a subject of active research in monetary economics and finance. Hanson and Stein (2015) and Nakamura and Steinsson (2018) showed that change in Fed Funds rate affects bond yields for maturities up to 10 years. For the first 5 years such effect is almost one-to-one. In other words, when the Fed surprisingly increases short-term rate, yields on Treasury notes at maturities between 2 and 5 years go up by almost the same amount.
Two main explanations for such effect are changes in expected short-term rate in the future and a change in term premium. Hanson and Stein (2015) argue that such reaction is mostly due to a change in risk premium/term premium, while Nakamura and Steinsson (2018) advocate for expectation of short-term rate in the future as the main channel. Distinguishing between the two explanations empirically is pretty difficult, since neither of the two variables is directly observable.
Intuitively, change in Fed Funds rate could affect expected short-term rates in the future through revealing some information about the state of the economy. Usually the Fed raises the rate when the economy is doing well, so such action of the Fed could be seen as the good news about the future of the economy. Hence future short-term rate will go up.
Change in Fed Funds rate could affect risk premium via so-called reach for yield effect. Lower rate could mean increased risk taking, since many investors do not want to invest in bonds if their yields are very low (e.g., 0-1%).
This question is not settled and is a subject of very active research right now.