Context. The third Gaia Data Release (DR3) provided photometric time series of more than 2 million long-period variable (LPV) candidates. Anticipating the publication of full radial-velocity data planned with Data Release 4, this Focused Product Release (FPR) provides radial-velocity time series for a selection of LPV candidates with high-quality observations. Aims. We describe the production and content of the Gaia catalog of LPV radial-velocity time series, and the methods used to compute the variability parameters published as part of the Gaia FPR. Methods. Starting from the DR3 catalog of LPV candidates, we applied several filters to construct a sample of sources with high-quality radial-velocity measurements. We modeled their radial-velocity and photometric time series to derive their periods and amplitudes, and further refined the sample by requiring compatibility between the radial-velocity period and at least one of the G, G_BP, or G_RP photometric periods. Results. The catalog includes radial-velocity time series and variability parameters for 9614 sources in the magnitude range 6 ≲ G/mag ≲ 14, including a flagged top-quality subsample of 6093 stars whose radial-velocity periods are fully compatible with the values derived from the G, G_BP, and G_RP photometric time series. The radial-velocity time series contain a mean of 24 measurements per source taken unevenly over a duration of about three years. We identify the great majority of the sources (88%) as genuine LPV candidates, with about half of them showing a pulsation period and the other half displaying a long secondary period. The remaining 12% of the catalog consists of candidate ellipsoidal binaries. Quality checks against radial velocities available in the literature show excellent agreement. We provide some illustrative examples and cautionary remarks. Conclusions. The publication of radial-velocity time series for almost ten thousand LPV candidates constitutes, by far, the largest such database available to date in the literature. The availability of simultaneous photometric measurements gives a unique added value to the Gaia catalog.