Each of the original colonies and later states were independent of each other and had their own local governing structures. Probably the most well known of them was the House of Burgesses of Virginia where some of the most known Founding Fathers served (Washington, Jefferson, Madison, Henry, etc.).

In both cases the local governmental entities sent a delegation, although not specifically with instructions to create a declaration of independence or a constitution, but rather to attend this coalition of colonial (Declaration of Independence) or state (Constitution) representatives to discuss how to address the issues they faced. This was known as the Continental Congress.

In some cases the delegations were granted a large degree of authority such as that of Massachusetts when the Declaration of Independence was signed. Other states/colonies sent delegations with more of a observe and report assignment. In the case of South Carolina for example they did not vote for independence until after returning to South Carolina for further instruction from the local legislature.

The two processes are different from the standpoint that when the Declaration of Independence was signed they were completely independent colonies with ties due to geography. In the case of the Constitution they were now an officially united confederation of independent states. From the time of the Declaration of Independence (1776) until the ratification of the Constitution of the United States of America (1789) the States cooperated as a loose knit confederation with official recognition by several European powers.

So in the case of the Constitution you now had an officially recognized federal body. There was never an explicit mandate from the states to modify the Articles of Confederation into the Constitution it was more of an organic process that grew over time as many of the Founding Fathers saw the many weaknesses inherent in the federal structure of the time until there was a general consensus to attempt to first revise the articles and eventually deciding to start from scratch.