We consider \(Q||C_{\max }\) , the problem of scheduling n jobs on m uniform machines while minimizing the makespan, in an online setting with migration of jobs. In this online setting, the jobs are inserted or deleted over time, and at each step, the goal is to compute a near-optimal solution while reassigning some jobs, such that the overall processing time of reassigned jobs, called migration, is bounded by some factor \(\beta \) times the processing time of the job added or removed. We propose an Efficient Polynomial Time Approximation Schemes (EPTAS) with an additional load error of \(\mathcal {O}(\varepsilon p_{\max })\) and constant amortized migration factor \(\beta \) , where \(p_{\max }\) is the maximum processing time in the instance over all steps. As an intermediate step, we obtain an Efficient Parameterized Approximation Scheme (EPAS), a \((1+\varepsilon )\) -competitive algorithm parameterized by \(p_{\max }\) and the number of different processing times d in an instance, with \(\beta \) bounded in a function of \(p_{\max }\) , d and \(\varepsilon \) .