For years, something strange kept happening on Wall Street.

Before a big shareholder could carry out plans to sell a slug of stock, the price dropped. It was as if other investors knew what was coming.

These transactions, known as block trades, are supposed to be a secret between the selling shareholders and the investment banks they hire to execute the trades. But a Wall Street Journal analysis of nearly 400 such trades over three years indicates that information about the sales routinely leaks out ahead of time—a potentially illegal practice that costs those sellers millions of dollars and benefits banks and their hedge-fund clients.