Tax competition is a double-edged sword: it keeps government on their toes regarding their expenses, but it also saps their ability to provide public goods. Generally, you would want some level of cooperation across authorities, like in a cartel, as pure tax competition is not believed to reach optimal outcomes. This is particularly true with environmental regulation and taxation.
Cees Withagen and Alex Halsema claim that tax competition may lead to a first best in an environment with cross-border pollution. The important word here is "may" as it still needs to be established that it can happen in practice. The innovation to their approach is to include endogenous capital to the model: capital can flee to other jurisdictions if it is taxed too high. Governments then play against each other. This pushes them to keep capital tax rates low. With more capital and income, demand for environmental quality is higher and emission tax rates are higher. With some luck, the latter may get exactly at the first best. It all depends on the environmental demand parameters, which are hard to estimate, though. A good opportunity to find some better measurements.