We demonstrate temporal pulse compression in gas-filled kagomé hollow-core photonic crystal fiber (PCF) using two different approaches: fiber-mirror compression based on self-phase modulation under normal dispersion, and soliton effect self-compression under anomalous dispersion with a decreasing pressure gradient. In the first, efficient compression to near-transform-limited pulses from 103 to 10.6 fs was achieved at output energies of 10.3 μJ. In the second, compression from 24 to 6.8 fs was achieved at output energies of 6.6 μJ, also with near-transform-limited pulse shapes. The results illustrate the potential of kagomé-PCF for postprocessing the output of fiber lasers. We also show that, using a negative pressure gradient, ultrashort pulses can be delivered directly into vacuum.