Here are two handy homemade tools for the lathe spindle.
On the top is a knockout bar made from a piece of non-marring closet rod. It's used to rap the back end of a center stuck in the spindle taper. It has an aluminum "bumper" to prevent damage to the center. The chunk of scrap steel on the left end is just there to provide some heft so I don't need to use a hammer.
On the bottom is a bottle brush from the kitchen supply store fitted with a handle made from select broomstick aged for twenty years in my shop, Garaj Mahal. It's a wonderful tool for getting swarf out of the spindle. With the long handle I can push it all the way through the spindle and deposit all the swarf on the intricate inlaid (with grease stains) Garaj floor.
If you're one of those lucky guys with a great yawning chasm in the spindle, check out the supply of toilet brushes. Some are cylindrical and about 3" in diameter.