Tests of local position invariance (LPI) made by comparing the relative redshift of atomic clocks based on different atoms have been carried out for a variety of pairs of atomic species. In most cases, several absolute frequency measurements per year are used to look for an annual signal, resulting in tests that can span on order of a decade. By using the output of continuously running clocks, we carry out LPI tests with comparable or higher precision after less than 1.5 years. These include new measurements of the difference in redshift anomalies beta for hyperfine transitions in 87Rb and 133Cs and in 1H and 133Cs and a measurement comparing 87Rb and 1H, resulting in a stringent limit on LPI, beta(sub Rb) - beta(sub H) = (-2.7 +/- 4.9) x 10 to the -7th power. The method of making these measurements for continuous clocks is discussed.