diff --git a/src/sch/Sch.jl b/src/sch/Sch.jl
index 82cedbbe2..88ab68dbd 100644
--- a/src/sch/Sch.jl
+++ b/src/sch/Sch.jl
@@ -25,7 +25,8 @@ Fields
 - ready::Vector{Thunk} - The list of `Thunk`s that are ready to execute
 - cache::Dict{Thunk, Any} - Maps from a finished `Thunk` to it's cached result, often a DRef
 - running::Set{Thunk} - The set of currently-running `Thunk`s
-- thunk_dict::Dict{Int, Any} - Maps from thunk IDs to a `Thunk`
+- running_worker::Dict{Thunk,Int} - Map from running `Thunk` to its worker
+- thunk_dict::Dict{Int, Thunk} - Maps from thunk IDs to a `Thunk`
 - node_order::Any - Function that returns the order of a thunk
 - worker_pressure::Dict{Int,Int} - Cache of worker pressure
 - worker_capacity::Dict{Int,Int} - Maps from worker ID to capacity
@@ -44,7 +45,8 @@ struct ComputeState
     ready::Vector{Thunk}
     cache::Dict{Thunk, Any}
     running::Set{Thunk}
-    thunk_dict::Dict{Int, Any}
+    running_worker::Dict{Thunk, Int}
+    thunk_dict::Dict{Int, Thunk}
     node_order::Any
     worker_pressure::Dict{Int,Int}
     worker_capacity::Dict{Int,Int}
@@ -345,8 +347,34 @@ function schedule!(ctx, state, procs=procs_to_use(ctx))
                 pop_and_fire!(ctx, state, p)
             end
         end
+        schedule_memory!(ctx, state)
     end
 end
+
+function schedule_memory!(ctx, state)
+    sent = Set{Thunk}() # TODO: Store this in state
+    for up in state.running
+        proc = state.running_worker[up]
+        for down in state.dependents[up]
+            down in sent && continue
+            push!(sent, down)
+            for (idx,input) in filter(x->!istask(x[2]), collect(enumerate(down.inputs)))
+                remote_do(prefetch, proc, down.id, idx, Dagger.tochunk(input))
+            end
+        end
+    end
+end
+
+const CACHED_INPUTS = Dict{Int,Dict{Int,Any}}()
+
+function prefetch(tid, idx, chunk)
+    d = get!(()->Dict{Int,Any}(), CACHED_INPUTS, tid)
+    if !haskey(d, idx)
+        d[idx] = move(OSProc(), chunk)
+    end
+    nothing
+end
+
 function pop_and_fire!(ctx, state, proc)
     task = pop_with_affinity!(ctx, state.ready, proc)
     if task !== nothing
@@ -421,6 +449,7 @@ end
 
 function fire_task!(ctx, thunk, proc, state)
     push!(state.running, thunk)
+    state.running_worker[thunk] = proc.pid
     if thunk.cache && thunk.cache_ref !== nothing
         # the result might be already cached
         data = unrelease(thunk.cache_ref) # ask worker to keep the data around
@@ -469,6 +498,7 @@ end
 
 function finish_task!(state, node, thunk_failed; free=true)
     pop!(state.running, node)
+    delete!(state.running_worker, node)
     if !thunk_failed
         push!(state.finished, node)
     else
@@ -526,6 +556,7 @@ function start_state(deps::Dict, node_order, chan)
                   Vector{Thunk}(undef, 0),
                   Dict{Thunk, Any}(),
                   Set{Thunk}(),
+                  Dict{Thunk, Int}(),
                   Dict{Int, Thunk}(),
                   node_order,
                   Dict{Int,Int}(),
@@ -568,7 +599,12 @@ end
         fetch.(map(Iterators.zip(data,ids)) do (x, id)
             @async begin
                 @dbg timespan_start(ctx, :move, (thunk_id, id), (f, id))
-                x = move(to_proc, x)
+                x = if haskey(CACHED_INPUTS, thunk_id) &&
+                       haskey(CACHED_INPUTS[thunk_id], abs(id))
+                    CACHED_INPUTS[thunk_id][abs(id)]
+                else
+                    move(to_proc, x)
+                end
                 @dbg timespan_end(ctx, :move, (thunk_id, id), (f, id))
                 return x
             end
@@ -592,6 +628,9 @@ end
         bt = catch_backtrace()
         RemoteException(myid(), CapturedException(ex, bt))
     end
+    if haskey(CACHED_INPUTS, thunk_id)
+        delete!(CACHED_INPUTS, thunk_id)
+    end
     @dbg timespan_end(ctx, :compute, thunk_id, (f, to_proc, typeof(res), sizeof(res)))
     metadata = (pressure=ACTIVE_TASKS[],)
     (result_meta, metadata)