Some countries currently or historically have given immigrants voting rights. Undue immigrant influence over domestic politics (sometimes referred to as “political externalities“) is a common reason given to restrict immigration, and suffrage generally tends to magnify the influence of those immigrants given the vote.
Keyhole solutions such as restrictions on acquisition of citizenship by immigrants are a common proposal to address concerns about immigrant influence. However, some express concern that excluding immigrants from the franchise or citizenship will not be sustainable in the long run or under reasonably-expectable conditions.
The experience of countries who have granted (or taken back) immigrant access to the ballot box may prove to be instructive for both sides of the argument here.
For a survey of which countries historically or currently grant immigrants some level of suffrage, see the paper Noncitizen Voting Rights: A Survey of an Emerging Democratic Norm by David C. Earnest (2003). Earnest also authored a comparative overview of different countries’ experiences with non-citizen suffrage: Voting Rights for Resident Aliens: A Comparison of 25 Democracies (2003).