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We address the nature of the giant clumps in high-z galaxies that undergo violent disc instability, distinguishing between long-lived and short-lived clumps. We study the evolution of long-lived clumps during migration through the disc via an analytical model tested by simulations and confront theory with CANDELS-HST observations. The clump ‘bathtub’ model, which considers gas and stellar gain and loss, involves four parameters: the accretion efficiency α, the star formation rate (SFR) efficiency ϵd, and the outflow mass-loading factors for gas and stars, η and ηs. The corresponding time-scales are comparable to the migration time, two-three orbital times. The accretion-rate dependence on clump mass, gas, and stars, allows an analytical solution involving exponential growing and decaying modes. For the fiducial parameter values there is a main evolution phase where the SFR and gas mass are constant and the stellar mass is rising linearly with time. This makes the inverse specific SFR an observable proxy for clump age. When η or ϵd are high, or α is low, the decaying mode induces a decline of SFR and gas mass till the migration ends. Later, the masses and SFR approach an hypothetical exponential growth with a constant specific SFR. The model matches simulations with different, moderate feedbacks, both in isolated and cosmological settings. The observed clumps agree with our predictions, indicating that the massive clumps are long-lived and migrating. A challenge is to model feedback that is non-disruptive in massive clumps but suppresses SFR to match the galactic stellar-to-halo mass ratio.