When, by any change of county boundaries made by the Legislature of this State, any district is transferred in whole or in part to another county than the one within which it was organized, then as to such part of the district so transferred to another county, the same power and authority shall exist, and the same proceedings shall be had in the conduct and management of the district, and in the levy and collection of assessments, and in all other ways with respect to its conduct and management, as though it had been originally organized in the county where it was organized and partially in the county to which it is transferred.