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In the strongly interacting limit of the Hubbard model localized double-occupancies form effective hard-core bosonic excitations, called a doublons, which are long-lived due to energy conservation. Using time-dependent density-matrix renormalisation group we investigate numerically the dynamics of doublons arising from the sudden expansion of a spatially confined band-insulating state in one spatial dimension. By analysing the occupation scaling of the natural orbitals within the many-body state, we show that doublons dynamically quasicondense at the band edges, consistent with the spontaneous emergence of an eta-quasicondensate. Building on this, we study the effect of periodically driving the system during the expansion. Floquet analysis reveals that doublon-hopping and doublon-repulsion are strongly renormalised by the drive, breaking the eta-SU(2) symmetry of the Hubbard model. Numerical simulation of the driven expansion dynamics demonstrate that the momentum in which doublons quasicondense can be controlled by the driving amplitude. These results point to new pathways for engineering non-equilibrium condensates in fermionic cold-atom experiments and are potentially relevant to driven solid-state systems.