Context. Central compact objects (CCOs) are a peculiar class of neutron stars, primarily encountered close to the center of young supernova remnants (SNRs) and characterized by thermal X-ray emission. Aims. Our goal is to perform a systematic study of the proper motion of all known CCOs with appropriate data available. In addition, we aim to measure the expansion of three SNRs within our sample to obtain a direct handle on their kinematics and age. Methods. We analyze multiple archival Chandra data sets, consisting of HRC and ACIS observations separated by temporal baselines between 8 and 15 years. In order to correct for systematic astrometric uncertainties, we establish a reference frame using X-ray detected sources in Gaia DR2, to provide accurate proper motion estimates for our target CCOs. Complementarily, we use our coaligned data sets to trace the expansion of three SNRs by directly measuring the spatial offset of various filaments and ejecta clumps between different epochs. Results. In total, we present new proper motion measurements for six CCOs, among which we do not find any indication of a hypervelocity object. We tentatively identify direct signatures of expansion for the SNRs G15.9+0.2 and Kes 79, at estimated significance of $2.5sigma$ and $2sigma$, respectively. Moreover, we confirm recent results by Borkowski et al., measuring the rapid expansion of G350.1$-$0.3 at almost $6000,{rm km,s^{-1}}$, which places its maximal age at $600-700$ years. The observed expansion, combined with the rather small proper motion of its CCO, implies the need for a very inhomogeneous circumstellar medium to explain the highly asymmetric appearance of the SNR. Finally, for the SNR RX J1713.7$-$3946, we combine previously published expansion measurements with our measurement of the CCOs proper motion to obtain a constraining upper limit of $1700$ years on the systems age.
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