How to present longitudinal data to LSTM for multiclass prediction
Use a LSTM, where the input is a sequence, specifically the sequence of records for that patient.  You can train the LSTM to predict anything you want -- e.g., the next step.  In LSTMs, the sequence normally is fixed-length (the same length for all people; you pad shorter sequences); each element of the sequence is a feature vector of same length (the same length for each element in the sequence.)  Each person in your dataset gives you at least one training instance for the LSTM.
It's beyond the scope of this question how to encode inputs, but you can read about standard methods; it's not specific to a LSTM.  For instance, you can encode categorical variables with a one-hot vector, and continuous variables directly.  Similarly for outputs, but you'll need to choose an appropriate loss function (e.g., cross-entropy loss for categorical variables, MSE for continuous values, etc.).  None of that is specific to LSTMs, so read up on how that is done with neural networks and the same methods will apply.
You'll probably need a lot of data to get the LSTM to work well.