Regression- and scale-invariant $M$-test procedures for detection of structural changes in linear regression model was developed and their limit behavior under the null hypothesis was studied [M. Hušková: Some invariant test procedures for detection of structural changes. Kybernetika 36 (2000), 401-414.]. In the present paper the limit behavior under local alternatives is studied. More precisely, it is shown that under local alternatives the considered test statistics have asymptotically normal distribution.