Every grant, deed, conveyance, contract to purchase or lease made since June 15, 1911, or hereafter made to any person, firm, or corporation, for a right-of-way for a private railroad, skid road, canal, flume, watercourse, or other easement, over or across any public lands for the purpose of, and to be used in, transporting and moving timber, minerals, stone, sand, gravel, or other valuable materials of the land, shall be subject to the right of the state, or any grantee or lessee thereof, or other person who has acquired since June 15, 1911, or shall hereafter acquire, any lands containing valuable materials contiguous to, or in proximity to, such right-of-way, or who has so acquired or shall hereafter acquire such valuable materials situated upon public lands or contiguous to, or in proximity to, such right-of-way, of having such valuable materials transported or moved over such private railroad, skid road, flume, canal, watercourse, or other easement, after the same is or has been put in operation, upon paying therefor just and reasonable rates for transportation, or for the use of such private railroad, skid road, flume, canal, watercourse, or other easement, and upon complying with just, reasonable and proper rules and regulations relating to such transportation or use, which rates, rules, and regulations, shall be under the supervision and control of the utilities and transportation commission.