The Fubini numbers count the permutations of horse racing where ties are possible. The closely related r-horse numbers count the finishes of a horse race where some subset of r horses agree to finish the race in a specific relative strong ordering. We express the r-Fubini numbers as a sum of r index-shifted sequences of Fubini numbers weighted with the signed Stirling numbers of the first kind. We use a novel shift operator counting argument. Further, we demonstrate the eventual modular periodicity of r-Fubini numbers. Their maximum period is determined to be the Carmichael function of the modulus. The maximum period occurs in the case of an odd modulus for Fubini numbers.