Economic convergence of the new member states (NMS) of the EU towards the old EU countries (EU-15), not only in terms of real income, but also in nominal terms, is of paramount importance for the whole of the EU. We build a dynamic CGE model, starting from the Balassa-Samuelson two-sector framework, but modify and enlarge it with forward-looking investment, consumption, and labour mobility behaviour to address several other issues like welfare and sustainability in terms of foreign indebtedness. At the same time we evaluate the impact of convergence on the EU-15 countries also, by endogenising offshoring and the related FDI flows from them to the NMS. Thereby we identify various effects of relocation and globalisation on the EU-15 enlarging the standard set of effects of globalisation and demonstrate the key role of their dynamic nature in the process of convergence. We find that in a general equilibrium setting fears of large adverse effects of a relocation of EU-15 manufacturing to the NMS are not well founded. In contrast, offshoring appears to be a win-win case for both the EU-15 and the NMS in terms of real income. The convergence of the NMS is fairly rapid, but will involve a persistent rapid inflation rate.