In a medium saucepan heat oil over medium heat. Add onion, ginger, and garlic; cook until onion is tender but not brown.
Stir sugar, vinegar, lemon juice, mustard, allspice, and cloves into onion mixture. Bring to boiling, stirring to dissolve sugar. Reduce heat. Simmer, uncovered, for 5 minutes. Stir in peaches and cherries. Return to boiling; reduce heat. Simmer, uncovered, about 10 minutes or until mixture is thickened, stirring occasionally.
Ladle hot chutney into hot sterilized half-pint canning jars, leaving a 1/2-inch headspace. Wipe jar rims; adjust lids and screw bands. Process filled jars in a boiling-water canner for 10 minutes (start timing when water returns to boiling). Remove jars from canner; cool on wire racks. (Or, ladle into half-pint freezer containers, leaving a 1/2-inch headspace. Seal and label. Store the chutney in the refrigerator for up to 2 weeks or in the freezer for up to 6 months.)