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The standard procedure for single-booking connecting flights with a single layover in the US goes like this:
You check-in a your point of departure. You should get both boarding passes, and your bags should be tagged through to your final destination.
When you land, you go through immigration (passport control, operated by CBP), reclaim your bags, and go through customs (CBP again).
Right after customs, you should have a counter to drop your bags again
You go to the terminal for your second flight, where you go through security (TSA), and get to your gate.
There are some situations where you won't be able to get your boarding pass for the second flight, but as far as I know, this is mostly in situations where there's a long layover, and check-in is not yet open for the second flight when you check-in for the first one. In that case, you usually get your second boarding pass from a transfer desk or directly from the gate agent.
Note that this works only for flights booked on a single ticket (same e-ticket number, same PNR) for airlines that have "interlining" agreements, which is the case of Virgin Atlantic and Air New Zealand.
Upvote:-1
I don't have hard info on this but currently I have only heard of issues with Miami and Atlanta. I have seen some comments saying that LAX is OK (but LAX can be crazy on a "normal" day) but my gut feeling is that the situation is fluid (and so far off the charts that it is almost incomprehensible) and as such it is hard to predict what will happen in a weeks time.
I don't fancy being you getting off a flight from Heathrow and then immediately jumping on one to NZ but there will be nothing you can do except "grin and bare it". However you will be departing from Tom Bradley (which is dedicated to international flights) so the TSA staff there will be aware that you can't just easily jump on the next flight. Worst case scenario is that you get stuck in LA for a night and rebooked on the next Air NZ flight.
I do expect that you will get both sets of boarding passes in London (but I don't have proof - I only fly in and out of LAX domestically, and get both sets when initially I check in) so I believe that you should be able to simply hop over to TBIT
You are taking a well trodden path with your Virgin Atlantic flights so my final bit of advice is simply to ask Virgin staff when you hit the ground in LAX - they will have the best/current information for your situation at the time.