I had one of my agents call from out of town today and ask me to take an evening to show one of her buyers some properties. I am not interested in filling in for my agents and don't want to start a habit of it. I told her "no", that she would need to call one of our other agents and make an arrangement with them. She argued that I would make money off of the sale when it closes. I explained that I would be happy to assist with training, mentoring, helping her write her first few offers and listing contracts, etc, but I would not fill in for her while she is out of town.
How is this handled at other offices? Do you just fill and and take an extra split, tell her "no", or do it for free?

Whenever I have to do this (and I did it twice this weekend) I charge a showing fee if the buyer writes an offer and it goes to closing. Usually my fee is $300. This is to compensate me for my time.

The argument that "I'm going to make money off it anyway" is hogwash. I have plenty else to do and if I'm called away from those duties, then everything else suffers. I don't do this to be arrogant or anything, but time is money.

Usually agents cover for each other on vacations and such. They either split the affected side or pay out a referral fee. I don't care as long as it's covered.

Sometime I show for free, but I don't make a habit out of it.

I prefer that they have another agent cover for them if possible, but at the end of the day I consider all of the clients to be my clients, I trust the agents to serve them as I would myself.

I don't make money off the agents, they contribute to the company and the company pays them for their work, just like in any other profitable business.

I see the word "team" being tossed around a lot, but in my office we work as one.

What does your office policy say?
It should address exactly this type of situation. In the absence of a policy, I'd expect my broker to pick up the slack or designate someone to help out, with the understanding that there will be a payback.
One office I worked in suggested to work/trade duties with another agent, or pay them.
I have paid other agents to attend my closings when I was unavailable.

The agent should have rescheduled the showings for the next day. The clients would have understood. It is not your responsibility to show properties for your agents. Word gets out you are willing to do that and others will follow whenever something comes up.