Quantitative localization and comparison of invariant distances of domains in $\mathbb C^n$

We obtain explicit bounds on the difference and ratio between "local" and "global" Kobayashi distances in a domain of $\mathbb C^n$ as the points go toward a boundary point with appropriate geometric properties. We use this for the global comparison of various invariant distances. We provide some sharp estimates in dimension $1$.
Comment: v2: Results about ratio of invariant distances and some references have been added, some proofs have been streamlined. v3: Major revision. To appear in J. Geom. Anal