The mobility of alkaline earth ions, Mg+, Ca+, Sr+ and Ba+, in liquid xenon is measured for the first time. The mobility of Tl+ is also determined for comparison with a measurement by other researchers. The Atkins cluster model for positive ions in non-polar liquids, based on the electrostriction effect, gives general agreement with the magnitude of the mobility values. This is some evidence that the positive ions form a snowball rather than a bubble structure in liquid xenon. However, the temperature dependence of the mobility does not match well with the Atkins theory, so there are still open theoretical questions on the nature of the environment of alkaline earth ions in liquid xenon. The lower mobility of Mg+ and Ba+ may be explained by a better size match to interstitial and substitutional sites, respectively, in solid Xe. These measurements are motivated by the development of a new technique to search for neutrino masses through 0νββ decay of 136Xe. A key component of one version of the proposed experiment is tagging of 136Ba+ daughter ions in liquid 136Xe by laser-induced fluorescence.