The Treaty of Lisbon requires that EU member-states have an executive responsible to a directly-elected Parliament.

The EU has a directly elected Parliament.

The EU has a slightly complicated executive. There is the Council of Ministers, which proposes legislation and which appoints the Commission. The Council is made up of democratically elected politicians (notably, politicians who do not have a direct personal mandate -- appointees to upper houses, for example, are generally excluded from direct personal participation in the Council). The Commission, which generally implements legislation passed by the Council and the Parliament, is chosen by the Council of Ministers and must be approved by the Parliament.

So the Commissioners are somewhat remote from the voting, but are responsible to two sets of directly elected politicians: those forming national executives, and those directly elected to the European Parliament.

There are plans to have the equivalents of Commissioners directly elected on a pan-European basis, however this requires unanimity among the 28 member-states, and that has not yet been achieved.

There are also plans to have more proportional national-parliamentary representation in a future version of the Council. In part this is because there are broad coalitions and complex national structures in a few member-states. Already there is an expectation that ministers show that they speak for the majority of their national parliaments on key matters (e.g. in formal QMV settings). These rules were adopted by elected politicians in the 28 national governments.

The Parliament suffers from MEPs not being very close to their voters -- the electorates are larger (much larger in some proportional representation cases) and they have less direct influence over purely local and personal matters than national or regional politicians. But they still campaign, and some MEPs are better than others at being close to their electorates.

The Commission has the disadvantage of being responsible to two different bodies: the Council and the Parliament, either of which can remove them. For a variety of reasons, a huge number of MEPs represent non-governing parties strongly opposed to some of their national governments' goals. Commissioners are also often former politicians in the party "opposite" the one in power in their home nation-state. (This is an alternate source of pressure by elected politicians to make sure the Council reflects something broader than the narrow interests of the various governments of the day, to the extent that those are mainly aligned party-wise or philosophically, which is not the case on numerous issues).

Freedom of movement within the EU is something supported by the vast majority of elected politicians across Europe -- in the national parliaments, in the national governments that those parliaments directly choose or support, in the European Council (which is, again, those same national governments), and in the European Parliament (directly elected). Criticism of Freedom of Movement should not really centre on being somehow undemocratic, or at least not more undemocratic than all the member states themselves.

Returning to your question, the Treaty of the European Union requires member-states to guarantee your access to your MEP (representing you where you live within the EU) and to local elected officials (again, where you live) as well as to elected members of the national parliaments in the member-state(s) of your nationality (wherever you happen to live). This is a very active area of European law (there are some cases before the ECJ where the Commission and others are supporting nationals of member-states they do not live in with respect to their right to access to members of parliaments of those member-states) and generally taken seriously.

So the EU is a guarantor of your right to complain to several democratically elected officials about their support for free movement of labour within the EU, and to vote on that basis if you wish (e.g., during EP elections or elections in your home state(s)).