Questions about Customs/Insurance/Transport for your equipment?
I can only write from the experience of travelling from the UK to many other countries with kit a few years ago (pre-covid era):
In the UK, if you provide an equipment list to the airport customs desk before you leave, they will examine it, take a copy and stamp it.  The form is called a Returned Goods Relief Form.  This is to prove that you are taking valuable equipment out of the country for filming purposes, and plan to return with it (not sell it overseas and avoid paying taxes).
The list needs to show every item of equipment you are carrying and should have columns for: Manufacturer, Manufacturers Model, Serial Number, Value (£).
Print many copies of the equipment list, so that any official who demands one can take one.
Put a copy of each equipment list in each bag, Put a &quot;lost luggage&quot; note in each suitcase in the local language, and English.  
Customs in the incoming country is another issue.  It depends on the country.  Research it carefully. Some countries need you to buy a &quot;carnet&quot;.
Make sure no bag weighs more that the allowed limit by the airline.
With regards to batteries, the correct procedure a few years ago was to inform the airline that you would be carrying batteries. Make sure the weight of the batteries did not exceed their rules.  If they are Lithium Polymer (LiPo) batteries (often used in drones), carry them in LiPo bags which should reduce fire risk if they malfunction on the journey. Any exposed terminals should be taped over.  
The airlines usually (and correctly) state that you should carry the batteries in hand luggage, so if they do malfunction, it would quickly become apparent - if they were in the hold, it would be too late to do anything.  Security guards in airports often aren't aware of this rule, and may try to get you to put batteries in the hold.  Allow time for this when you travel.
Officials love letters. Get a letter from the airline confirming that you can travel with the batteries. Take a letter from your employer confirming that you will be filming in x country for y days and address it &quot;to whom it may concern&quot;.  Print many copies on headed paper. 
Can't comment on insurance as it was handled by the company I worked for.  Large broadcasters usually have insurance, but often the excess is so high it is effectively useless. More important is the third party liability insurance, in case you injure somebody else during your work.
Also - use Peli cases. They're waterproof, fairly indestructible, and if they do get broken, they will offer you a replacement.