Standard triggers at the Large Hadron Collider are designed to select events with high momentum Standard Model particles which originate from the proton-proton collision of interest. However, several Beyond the Standard Model scenarios predict signatures with displaced or extremely low momentum charged particles, and pose significant challenges for LHC triggers. This article presents a study of track-based triggers for longlived particles and other unconventional exotic signatures at the upcoming High Luminosity LHC. Representative scenarios studied include soft-unclustered-energy-patterns resulting in large multiplicities of low momentum tracks, a GMSB Supersymmetry scenario with longlived staus resulting in displaced leptons or anomalous prompt tracks, and a higgs portal scenario with long-lived scalars resulting in displaced hadronic tracks. Trigger efficiency is measured as a function of the baseline parameters of a track trigger, including transverse momentum and impact parameters. Recommendations for future hardware-based track triggers are presented.