I recall when I played in a supper club trio back in the late 70s, going from the US into Canada through Customs, they would make a list of all our equipment with serial number. We had to leave a deposit that represented a certain percentage of the total value of the equipment. Coming back across the border, we had to go through the same Customs point to get rechecked. If any of the equipment was missing, they assumed it was sold and therefore kept the deposit. If all the equipment on their list was still in our possession, we got the deposit back. As I recall, it was equally a hassle for Canadian musicians to come to the US and later return to Canada. I have no idea what issues there might be for musicians in this situation today. I got out of that line of work and moved on, not because of these issues, but the constant touring lifestyle just wasn't for me.
Later, in the 90s, I had to bring some computer boards into England. The Customs folks were having a strike, so it took a LONG time to get the stuff through Customs. There were British attorneys (called "Solicitors") at the airport who specialized in these kinds of things just waiting to help us through these issues.
Going across borders just seems to have its issues. I can't imagine what it must be like having the excitement of a new guitar purchase and then the worry about the instrument making it safely through these kinds of issues.