A nontaxable exchange would generally occur only if the mutual fund company moved the funds, say, either because one fund was discontinued and merged with or replaced another or requirements were met that allow the mutual fund company to reclassify the shares to shares with a lower expense ratio, such as from class C to class A in the same fund.
If you simply asked for shares to be sold and you then used the proceeds to purchase shares of a different fund, that's a taxable sale and any amount above your cost basis in the original shares would be taxable as either short- or long-term capital gains (or a combination of the two). In this case the brokerage will report the sale on Form 1099-B to be reported on your individual tax return. The cost basis of your new shares would generally be the purchase price of the new shares.
If you instead just moved the shares in-kind from one brokerage to another, with no actual sale and repurchase, this would not be a taxable transaction.
