Old Princeton’s Samuel Miller wrote about the role REs ought to play in the Church’s courts and his thoughts ought to be read at length:
The functions of the Ruling Elder are not confined to the congregation of which he is one of the rulers. It is his duty at such times, an in such order as as the constitution of the Church requires, to take his seat in the higher judicatories of the Church, and there to exercise his official share of counsel and authority. In every Presbytery, Synod and General Assembly of the Presbyterian Church, at least as many Ruling as Teaching Elders are entitled to a place; and in all the former, as well as the latter, have an opportunity of exerting an important influence in the great concerns of Zion. Every congregation, whether provided with a Pastor or vacant, is entitled, besides the Pastor, (where there is one,) to be represented by one Ruling Elder, in all meetings of the Presbytery and Synod; and in those bodies, vacant congregations, and those which are supplied with Pastors, are equally represented, each by an Elder, it is manifest that, if the theory of our ecclesiastical constitution be carried into effect, there will always be a greater number of Ruling Elders than of Pastors present. In the General Assembly, according to our constitutional plan, the numbers of each are precisely equal.
In these several Judicatories the Ruling Eider has an equal vote, and the same power, in every respect, with the Pastors. He has the same privilege of originating plans and measures, and of carrying them, provided he can induce a majority of the body to concur in his views; and thus may become the means of imparting his impressions, and producing an influence greatly beyond the particular congregation with which he is connected and, indeed, throughout the bounds of the Presbyterian Church in the United States. This consideration serves to place the nature and the importance of the office in the strongest light. He who bears it, has the interest of the Church, as a spiritual trust, as really and solemnly, though not in all respects to the same extent, committed to him as the Elder who, “labors in the word and doctrine.” He not only has it in his power, but is daily called, in the discharge of his official duties, to watch over, inspect, regulate, and edify the body of Christ: to enlighten the ignorant; to admonish the disorderly; to reconcile differences; to correct every moral irregularity and abuse within the bounds of his charge; and to labor without ceasing for the promotion of the cause of truth, piety, and universal righteousness in the Church to which he belongs, and wherever else he has an opportunity of raising his voice, and exerting an influence.
But when it is considered that those who bear the office in question are called upon, in their turn, to sit in the highest Judicatories of the Church; and there to take their part in deliberating and deciding on the most momentous questions which can arise in conducting ecclesiastical affairs; when we reflect that they are called to deliberate and decide on the conformity of doctrines to the word of God; to assist, as judges, in the trial of heretics, and every class of offenders against the purity of the Gospel; and to take care in their respective spheres that all the ordinances of Christ’s house preserved pure and entire; when, in a word, we recollect that they are ordained for the express purpose of overseeing and guarding the most precious concerns of the Church on earth; concerns which may have a bearing, not merely on the welfare of a single individual or congregation, but on the great interests of orthodoxy and piety among millions; we may surely conclude without hesitation, that the office which they sustain is one the importance of which can scarcely be over-rated; and that the estimate which is commonly made of its nature, duties and responsibility, is far, very far from being adequate.
You can read the rest of Dr. Miller’s essay here: http://www.reformed.org/books/ruling_elder/index.html
But suffice it to say, REs belong at General Assembly and Presbytery and they are called to be active and vigilant.