I've had to support several ESL graduate students in the past. In general, they were smart and capable but really struggled with the reading and writing aspects of graduate work. They did complete the prerequisite ESL courses and passed one of the standard English test to enter university studies (IELTS, TOEFL, TOEIC, etc) but there seems to be a disconnect between their preparation and their performance
I would like to know what other educators have done to support graduate students with limited English proficiency?
I think you should let the students know that they need additional practice and should be prepared to read more papers, seek help from the university Writing Center, and get a native English-speaking buddy to discuss the field with in-depth for practice. I know the last thing that any graduate student wants to do is more work, but it might help if you emphasize to them that you care about their long-term success in the field, and that reading and writing at a professional level in the field are crucial to that.
The foreign students and researchers I have worked with are primarily Chinese, so that colors my experience, I'm sure, but many of them were not educated with an eye to being good readers and writers. They may have been trained to read for memorization, not comprehension, because they expect to be told what to do by senior researchers rather than critically respond to ideas and generate their own. So it may not just be a language barrier you're dealing with here, but also a cultural barrier where research is conducted differently. Your students may not understand that they will be expected to interpret, rather than passively absorb, the literature, and that they will have to kick their reading skills up a notch if for no other reason than that.
See if you can enlist some of the native English-speaking grad students to reach out and lend a hand to their ESL colleagues. Again, I do understand that no grad student wants to do extra work (boy, DO I understand), but you can emphasize that it will benefit them, as well. They will improve their own understanding of the readings tremendously by having to explain them in terms their classmate can understand, and they will also become better collaborators. Make sure they know they can mention doing something like this in the "service" section of their CVs.