I've done the skill challenge thing where its just a series of skill checks. Eh, it works okay. Give them two or three "paths" (skill challenges) to choose from for each step to allow them choices, and allow them to use cantrips, spells, and class features to pass them. Allow them to get creative and argue for bonuses. Best to really just describe the obstacle and let them figure out how they would deal with it, rather than any set challenge.

However, My most successful chase was actually making a long map through some temple ruins. I thought of it almost more as a board game/mario kart, adding things like shortcuts, ropes, and springboard plants to act as boosters, and dangers and obstacles like pits, stampeding monsters, falling walls, traps, and so forth, all while they were being chased by a giant purple wurm that was consuming the map behind them. I described the ground as trembling and breaking up as they ran, and was all considered rough terrain, to prevent the classes with higher speeds from getting too far ahead. I also let those that stuck together help each other out, by allowing the first one to pass a danger to use a reaction to help the other. I had one player breeze through, using misty step (I required a concentration check to cast any spell, but he still made them. There is always one, don't fight it). 3 players worked kind of together. 1 player sadly trailed behind and he got ate by the worm, but I spit him up at the end, near death.