The gains of the right in the economic sphere - rollback of welfare state, privatisation, increase in rights, reach and power of corporations etc (basically the victory of capitalism, Thatcher/Regan and Neo-Liberalism) - squeezed most traditional left wing platforms and policies straight out of the Overton window, or at least the ones that offer a genuine chance of improving people's lives, Blair and the New humanitarianism did the same for the last vestiges of left wing foreign policy, diplomacy, disarmament, multilateralism, internationalism, UN as effective actor etc.
So what youre left with is social liberalism, shorn of any ideological framework and tradition, floating, isolated in a sea of neo-liberalism. There is no 'convergence' in the sense of left and right both arriving at these policies at the same time, rather, the right has taken the only policies of the left they can stomach (as they are 'soft' and offer no challenge to prevailing economic/market/military policies) and offered them as panacea to long suffering constituents, and the 'left' are fighting the same battle in order to disguise their own ideological bankruptcy and betrayal of their fundamental ideals.