It is known that two groups with an amalgamated subgroup can be embedded in a group, and if the given groups are finite, the embedding group can be chosen finite. The present paper deals with the question how `finite' can here be relaxed to `locally finite', `of finite exponent', or `periodic'. An example shows that two locally finite groups of finite exponent, with an amalgamated subgroup, may not be embeddable even in a periodic group. Conditions that ensure the possibility of such embeddings are then investigated. The principal tool is the `permutational product' of groups that has recently been introduced into the investigation of other embedding problems.