  Radiation reaction (RR) is the oldest still-unsolved problem in
electrodynamics. In addition to conceptual difficulties in its theoretical
formulation, the requirement of exceedingly large charge accelerations has thus
far prevented its unambiguous experimental identification. Here, we show how
measurable RR effects in a laser-electron interaction can be achieved through
the use of flying focus pulses (FFPs). By allowing the focus to
counterpropagate with respect to the pulse phase velocity, a FFP overcomes the
intrinsic limitation of a conventional laser Gaussian pulse (GP) that limits
its focus to a Rayleigh range. For an electron initially also
counterpropagating with respect to the pulse phase velocity, an extended
interaction length with the laser peak intensity is achieved in a FFP. As a
result, the same RR deceleration factors are obtained, but at FFP laser powers
orders of magnitude lower than for ultrashort GPs with the same energy. This
renders the proposed setup much more stable than those using GPs and allows for
more accurate \emph{in situ} diagnostics. Using the Landau-Lifshitz equation of
motion, we show numerically and analytically that the capability of emerging
laser systems to deliver focused FFPs will allow for a clear experimental
identification of RR.
