I've traveled around the US and to Europe with climbing gear. No chainsaws or rigging gear though.
One of THE BEST tips that I've ever read is to put a copy of an arbo supply catalog on the top of the pack or duffle bag. Maybe one from different companies. that way the bag checker might have an idea what all of this odd looking gear is for. If you have any carry on, even just a rope, include the catalogs. It might not be a bad idea to have some pics on your phone or camera of the gear in use too.
When I put the gear in the pack for checked luggage I put rope bags in the bottom. Next layer is my harness. Any loose biners and hard gear get clipped into a gob. When the bag checker grabs the stuff I don't want to have any loose gear skittering across the floor and getting lost.
throwbags get clipped together and to close to the top. I've seen the screen on the x-ray machine when my throwbags were on the bottom. The screener was amenable to having me explain what they were for instead of having to unload the whole pack. On the screen the look like h#nd gr#n#des
Flying with a used chainsaw is not likely. Camp stoves with attached fuel containers or even aluminum fuel bottles can't be brought on a plane. Best to ship them via UPS/FedEx, etc.
If my gear bag is close to shipping weight I'll bring a rope in my carry-on bag. It is the piece of gear that has the highest volume/density. Security can understand rope, they might not understand hardwear.
It doesn't take much gear to make the fifty pound weight limit. don't get too big a bag. Consider the extra weight from heavy duty rope bags etc. I've been one pound overweight and been told that 'something' had to go. My smart-mouth asked if I should tear off the covers to my books or pages out of magazines. The counter clerk was not amused
