The typical procedure when invited to give a talk at another university is for the invitee to get the plane ticket and then get reimbursed after the travel is completed. Lodging, meals, and other transportation costs are taken care of directly by the host institution.
I was recently thinking about all the professional travel expenses my husband and I currently have that have not yet been reimbursed. Including past travel that has yet to be reimbursed and future travel that won't be reimbursed until various times over the next 5-6 months, the sum is quite large at the moment: in the range of $7000. It is so large in part because the system has broken down a bit in terms of timely reimbursements for past travel relative to payments for future travel. In addition, because my teaching schedule is lighter in the spring term than in the fall, I scheduled most of my invited talks for next term. I just got a flock of plane tickets and loaded up my credit card.
For one talk I gave months ago but for which I have yet to be reimbursed, my hosts have been very apologetic but say that their departmental accountant has been deliberately losing receipts and taking a long time with all reimbursements. I have sent my receipt and social security number to the accountant 3 times so far. Methinks my department isn't the only one with a hostile zombie staff member. My hosts say that I am welcome to call their accountant and yell and/or whine, but that option doesn't appeal to me very much (yet).
The delay is annoying but not a huge problem for me. It occurred to me, though, that hostile zombie accountants could cause major problems for financially vulnerable people, such as some candidates for faculty positions (e.g., recent/current Ph.D. students interviewing at several/many places).
As long as there is a constant stream of reimbursements to offset the constant acquisition of new plane tickets, the system works pretty well. For me right now, the system is out of whack, with a drought at one end and a flood at the other. If things got dire, I would ask a host department to get my plane ticket for me, avoiding the reimbursement issue entirely. In the meantime, I will just try to enjoy the glorious adventure that is air travel in the U.S. these days, and not worry so much about $ matters.
9 months ago