Free, yes, to them. They usually do some basic fundraising, generally it comes from the congregation. They'll stand up at the beginning of service and ask for donations from people, and when they have enough they'll go.

As for work, I call it light work, because they aren't going down and building a mission for the people, they're usually doing something like painting or washing something. Things that locals are 100% capable of doing themselves, that is not really anything actual like building a well or anything. They do 1 or 2 days of this, and then the other 3 to 4 are on tours of the country, eating out, and basically being on vacation.

I know what you're trying to do, and I don't care. These "mission trips" are disingenuous to me. They aren't going down to help the people, they want to feel good about themselves and do the bare minimum to do so. They talk about how Jesus affected them personally to people who have never known the luxuries of life that we have, and then they jump on the bus back home, leaving the locals with a brand new bible and a fresh coat of paint, while patting themselves on the back for being such great christians. It's all a show for themselves and for the church.

Real actual use of that money could go to doctors without borders or UNICEF. There are actual groups doing actual good work in remote areas that desperately need help. These groups have the know how and the ability to do actual change for people, like building wells, providing medicine, building shelter, building schools, and more. They are the ones we should appreciate, they are the ones who should receive the money that was raised. I don't care how you try to word it, 20 teenagers for 3 days aren't going to make any meaningful impact on an impoverished community, actual doctors might though.

Edit, some resources on the actual problem of missionaries

A great video from someone who has experience missionaries coming directly, with excerpts from someone who went on a mission trip: Video

Article explaining how the lack of actual skills hurts more than helps, and all they're doing is building their own savior complex

People have died because missionaries offer fake care to the people, while raking in donations

In the end, modern missionaries is a well oiled tourism machine, bringing in a ton of money to small groups who help ferry missionaries over to these areas, give them a tour of the area, and send them home. The part that keeps the machine moving is the marketing and advertising arm of it, posting pictures of great work while not actually doing anything, but encouraging more 1st world white people to go.