In my opinion you would be much better off pre-shipping your equipment Fedex or UPS and picking up at the destination terminal. I don't even trust clients to receive my gear. May add an extra day or 1/2 day to your trip but giving the airlines access to your gear is just asking for trouble.
There is no doubt that moving equipment by air gets harder and harder and there is no secret, magic way to do it, other than preparation.
First I suggest getting a press card as depending on carrier you get 100lbs vs. 50lbs per case. Not all carriers recognize this, (continental out of JFK) but most will (American).
In europe it's just a crap shoot but flying from the states, it's much better to go non stop than transfer even if the airfare is higher.
The one bad experience I've had is flying to Charles DeGaulle and transferring on Air France to Barcelona. A work slowdown (strike) kept our equipment on the ground in Paris for 4 days. On return we drove from Barcelona to Paris to avoid the strike.
The one and only bad experience I've had with Fedex is one of our crew did not mark a case properly and the shipping label was not attached well, though we did finally get the case returned.
Given my choice I'd rather have my own equipment as we know it, it's paid for so the rental fees are mine and I can rely on it.
If the schedule permits and we're traveling domestically we ship to the hotel or production company/studio be Fedex ground This can save a lot and is just as reliable as standard Fedex.
If your shooting location only then find hotels that are production friendly. Obviously motel 6 won't be high on the list, but most hotels that cater to business are use to large shipments coming their way.
Even when we have the time luxury to ship early, cameras and a few essential lights/grip go with us on the plane so if anything goes wrong in shipping we can start the project or the casting.
I like cases filled to the max and heavy because I have a theory that big heavy stuff is a lot more difficult to steal than small light stuff.
In cities we are not based in I usually hire a gaffer who owns his own equipment and truck, if the project requires large grip like rollers, highboys, 12x frames, generators, etc. A independent gaffer will probably be the same price as
rolling your own using rental companies and gaffers that work in the cinema world are use to giving out detailed estimates that they will live to.
The also know their equipment have a reliable group of grips and swings for crew and know the area they're based in. A great gaffer can save you days of work, many hours in the day on production.
Even with a gaffer, prior to traveling we set up accounts with the rental companies, have maps to their locations and personal contacts. We always ask for cell phones so if we need something after hours.
We also prepare and file a carnet for long tours and the more prepared and official your are the easier it is. Most countries are fine with equipment because they know production brings money to their country, though some aren't.
Of all the places we work, the most difficult going in are Canada (almost impossible for an American without hiring a Canadian national that goes with you), Brazil and Moscow. Leaving the most difficult is South Korea and Moscow. The most unorganized is Italy.
I dig Moscow but everything is on slow motion, everything requires huge volumes of paperwork and bureaucracy.
The very easiest is Hong Kong. It's like come on in, spend some money, have some fun, see ya later. It's as easy as going NY to LA. In my experience Hong Kong is the best, very production friendly and they even repaired one of my 1ds3's in 3 hours.
The gaffer we work with in Hong Kong is the best in the world, the crew the hardest working and the most fun.
But the only rule of the road* is prepare for the worst, pray for the best and be backed up on everything.
*(at least the only rule I can mention on an open forum).
BC