I believe that an ecumenical council would be convened pretty much the same way that a local council is convened.
Consider: As of 1517, Roman Catholics counted 10 ecumenical councils: First Council of Nicaea (325), First Council of Constantinople (381), Council of Ephesus (431), Council of Chalcedon (451), Second Council of Constantinople (553), Third Council of Constantinople (680), Second Council of Nicaea (787), Fourth Council of Constantinople (869-870), Council of Florence (1431-1445), and Fifth Lateran Council (1512-1517).
But around the time of the Council of Trent (1545-1563), they decided that there had been another 8 ecumenical councils (which had hitherto been regarded as local): First Council of the Lateran (1123), Second Council of the Lateran (1139), Third Council of the Lateran (1179), Fourth Council of the Lateran (1215), First Council of Lyon (1245), Second Council of Lyon (1274), Council of Vienne (1311 –1312), and Council of Constance (1414 –1418).