Fill the joint with some form of flexible filler before you apply any caulking. The minimum width for an expansion joint is 1/4 inch, which is too wide for you to use caulking to fill the entire void. Instead, back-fill the joint with backer rod or spray foam until the joint is close to full. Leave only around 1/4 inch of depth for the caulking.
Cut the backer rod strips with a utility knife. Use a piece of wood, such as a stick, to push the backer rod into the expansion joint. Layer as many rods as you need until you reach the desired height within the joint. Alternatively, you can use spray foam to fill the bottom part of a joint, although it will expand so you need to ensure you put only enough to fill the joint to within 1/4 inch of the top.
Slice the tip off a tube of caulking with the utility knife. Insert the tube into the caulking gun. Press the tip of the tube into the expansion joint. Pull the trigger on the caulking gun and fill the expansion joint on top of the backer rod or foam. Fill the joint until it is flush with the top of the concrete, and work your way back along the joint until you finish the entire expansion joint.