For people hoping to get overseas latency better than fiber: not now, and probably never without a huge premium.
When they finally get inter-satellite comms, low-latency transport will probably be reserved for huge-premium tier accounts.
For everybody else, you can assume the satellites will aggressively dump packets to the nearest downlink terminal to feed into fiber. After they get laser links going, when there is no downlink terminal in view, packets might move to the next satellite ahead or behind on the same orbit, and then, if possible, down, or repeat. There will be no attempt to send packets to satellites in other orbits. They might sometimes delay traffic so as not to match performance of their premium tier, if it would otherwise be reliably fast.
For future super-high-premium tier accounts, packets will step through satellites to get to a terminal near the destination, and downlink there. Packets sent via this route will arrive whole milliseconds ahead of terrestrial fiber. Despite small amounts of traffic bouncing between orbit tracks, these accounts will constitute a substantial fraction of Starlink's total revenue.
In the eventual future, we can expect satellites to carry a few TB of on-board storage to proxy blockbuster media properties; and, to broadcast select real-time traffic, e.g. World Cup video. These will be done mainly to free up bandwidth.
Those are my predictions. If yours differ, explain.