1. Wipe them off with a clean cotton rag and water, removing all dirty spots, dried mud, dust and debris.
2. Wait until they dry.
3. Use a conditioner to moisturize the leather.
4. Buff with a clean rag; you can also brush at this point.
5. Apply whatever combinations of cream and/or wax polishes you prefer. Use colors that are lighter than the leather, if you wish to preserve the antique burnishing. Neutral polish is safe as well, but scuffs may remain visible with neutral polish, which some like as age/patina. Brush and buff with cotton rag between coats of polish.
7. Cover the boots with dust bags, and store them in a cool and dry place.
PS: avoid alcohol or saddle soap – these can be harsh on some leather. Consider using a toothbrush or other small welt brush to clean dirt along the sole edge.