Anion-exchanged compound - smaller 1H NMR integrations
On face value, there is nothing here that necessarily suggests a change in relaxation times. Before thinking too hard about what chemistry is happening, you should assure us that your data is reliable and consistent. You should provide details of your experimental conditions; are you using a calibrated &pi;/2 pulse for excitation or a shorter pulse approximation? What relaxation delay are you using? Have you measured T1 relaxations for these nuclei? You could consider repeating your experiment using a much greater relaxation rate between scans to see whether the integrations change.

These are more interesting phenomena in the spectra that need rationalising; loss of fine structure for downfield peaks for instance.

As for your question though, there are lots of examples of counterions changing relaxation times. This is a major tool used for probing outer sphere complexation, and can be used to determine how anions and cations interact around the ligand. It is used widely in protein structure (complexation with paramagnetic cations) and investigations with ionic liquids. If this is a counterion-induced relaxation change, you should be able to account for this by repeating your NMR experiment using appropriate parameters.