I am not sure if what I am about to relate is exactly what happens, but I watched several thermite welds being done a couple of years ago. The process involved two major components, One, a set of clamps and a crucible of sorts, with a funnel like lid, and the other a cardboard box that apparently contained the consumables; a pair of mold halves and a bag of thermite and maybe some other items, but I could not see all of it from my vantage point.
After removing the consumables from the box, they tossed it not too far away and then placed the mold halves on each side of where the joint was to be made and used the clamps to hold them in place. Then they sat the crucible on top and dumped a bag of thermite (I assume that is what it was) into the top of the crucible top and tossed the bag into the cardboard box.
After another man inspected the whole setup, then he tossed a match into the top of the crucible. It smoldered a few seconds and then gave off a large flame out the top with a shower of sparks. The space between the crucible and the rail smoked heavily and I could (from quite a distance) tell that something "flowed" from the crucible to the area below and a small amount of overflow went into a shallow pan they had hung from the side.
A minute or two after that they began to dismantle the whole apparatus, and after some banging on the shallow pan they used some tongs to toss its contents into the carboard box. The carboard box almost immediately began to smolder, and burst into flame, completely consuming the box, bag and all the other stuff they had tossed into it.
While the box was burning they knocked the mold pieces off, breaking them into lots of small rocks.
Then they attached a second machine to the rail a couple of feet away. This had an air operated grinding wheel on the end that was suspended directly over the place where the weld had been done. By pivoting the grinder where it was clamped to the rail, they swept the grinding wheel from side to side. It appeared that the grinding wheel would not go lower than the top of the surounding rail but I could not tell how that was controlled.
The total time to do all this was about 30 to 40 minutes and most of that time was spent in attaching and removing the equipment. The actual thermite burn was less than 45 seconds and the grinding time was no more than 2 or 3 minutes.
This was during the summer months. I also saw a similar operation in the winter time and it took comsiderably longer as they had to setup a set of gas burners on both sides of the rail for about 10-ft in both directions from the point where the weld was done (nearly continuous flame for the whole 20-ft on both sides with only a 2-ft gap in the middle).
They heated the rail for about 10 minutes before even starting to attach the crucible. This was done for two reasons. One is to get the rail warm such that the thermite weld would not chill too quickly and form a bad weld, and the other reason was to stretch the rail to the correct length to provide the proper tension for thermal expansion and contraction throughout the year. This was also near dusk and the thermite burn was absolutely spectacular! The whole joint glowed a deep red up until they started grinding on it.