diff --git a/tests/inference_on_device_test.py b/tests/inference_on_device_test.py
index a44a3d138..8570ac8d2 100644
--- a/tests/inference_on_device_test.py
+++ b/tests/inference_on_device_test.py
@@ -34,7 +34,7 @@
     prepare_for_sbi,
     check_embedding_net_device,
 )
-from sbi.utils.get_nn_models import posterior_nn
+from sbi.utils.get_nn_models import posterior_nn, likelihood_nn, classifier_nn
 
 
 @pytest.mark.slow
@@ -279,18 +279,20 @@ def test_train_with_different_data_and_training_device(
 
     inference = inference_method(
         prior,
-        density_estimator=(
-            "mdn_snpe_a"
-            if inference_method == SNPE_A
-            else "resnet"
+        **(
+            dict(classifier="resnet")
             if inference_method in [SNRE_A, SNRE_B]
-            else "maf"
+            else dict(
+                density_estimator=(
+                    "mdn_snpe_a" if inference_method == SNPE_A else "maf"
+                )
+            )
         ),
         show_progress_bars=False,
         device=training_device,
     )
 
-    theta, x = simulate_for_sbi(simulator, prior, 5)
+    theta, x = simulate_for_sbi(simulator, prior, 32)
     theta, x = theta.to(data_device), x.to(data_device)
     x_o = torch.zeros(x.shape[1])
     inference = inference.append_simulations(theta, x)
@@ -307,12 +309,12 @@ def test_train_with_different_data_and_training_device(
 
 
 @pytest.mark.gpu
-@pytest.mark.parametrize("inference_method", [SNPE_C])
+@pytest.mark.parametrize("inference_method", [SNPE_A, SNPE_C, SNRE_A, SNRE_B, SNLE])
 @pytest.mark.parametrize("prior_device", ("cpu", "cuda"))
 @pytest.mark.parametrize("embedding_net_device", ("cpu", "cuda"))
 @pytest.mark.parametrize("data_device", ("cpu", "cuda"))
 @pytest.mark.parametrize("training_device", ("cpu", "cuda"))
-def test_custom_density_estimator_training_device(
+def test_embedding_nets_integration_training_device(
     inference_method: NeuralInference,
     prior_device: str,
     embedding_net_device: str,
@@ -320,11 +322,11 @@ def test_custom_density_estimator_training_device(
     training_device: str,
 ) -> None:
 
-    # add warnings checks and other methods
+    # add other methods
 
     D_theta = 2
     D_x = 3
-    samples_per_round = 100
+    samples_per_round = 32
     num_rounds = 2
 
     x_o = torch.ones((1, D_x))
@@ -332,25 +334,73 @@ def test_custom_density_estimator_training_device(
     prior = utils.BoxUniform(
         low=-torch.ones((D_theta,)), high=torch.ones((D_theta,)), device=prior_device
     )
-    embedding_net = nn.Linear(in_features=D_x, out_features=2).to(embedding_net_device)
-    density_estimator = posterior_nn(model="maf", embedding_net=embedding_net)
 
-    with pytest.raises(Exception) if prior_device != training_device else nullcontext():
-        inference = inference_method(
-            prior=prior, density_estimator=density_estimator, device=training_device
+    if inference_method in [SNRE_A, SNRE_B]:
+        embedding_net_theta = nn.Linear(in_features=D_theta, out_features=2).to(
+            embedding_net_device
+        )
+        embedding_net_x = nn.Linear(in_features=D_x, out_features=2).to(
+            embedding_net_device
+        )
+        nn_kwargs = dict(
+            classifier=classifier_nn(
+                model="resnet",
+                embedding_net_x=embedding_net_x,
+                embedding_net_theta=embedding_net_theta,
+                hidden_features=4,
+            )
+        )
+    elif inference_method == SNLE:
+        embedding_net = nn.Linear(in_features=D_theta, out_features=2).to(
+            embedding_net_device
+        )
+        nn_kwargs = dict(
+            density_estimator=likelihood_nn(
+                model="maf",
+                embedding_net=embedding_net,
+                hidden_features=4,
+                num_transforms=2,
+            )
         )
+    else:
+        embedding_net = nn.Linear(in_features=D_x, out_features=2).to(
+            embedding_net_device
+        )
+        nn_kwargs = dict(
+            density_estimator=posterior_nn(
+                model="mdn_snpe_a" if inference_method == SNPE_A else "maf",
+                embedding_net=embedding_net,
+                hidden_features=4,
+                num_transforms=2,
+            )
+        )
+
+    with pytest.raises(Exception) if prior_device != training_device else nullcontext():
+        inference = inference_method(prior=prior, **nn_kwargs, device=training_device)
+
     if prior_device != training_device:
-        return
+        pytest.xfail("We do not correct the case of invalid prior device")
 
     theta = prior.sample((samples_per_round,)).to(data_device)
 
     proposal = prior
-    for _ in range(num_rounds):
-        X = MultivariateNormal(torch.zeros((D_x,)), torch.eye(D_x)).sample(
-            (samples_per_round,)
+    for round_idx in range(num_rounds):
+        X = (
+            MultivariateNormal(torch.zeros((D_x,)), torch.eye(D_x))
+            .sample((samples_per_round,))
+            .to(data_device)
         )
 
-        density_estimator_train = inference.append_simulations(theta, X).train()
+        with pytest.warns(
+            UserWarning
+        ) if data_device != training_device else nullcontext():
+            density_estimator_append = inference.append_simulations(theta, X)
+
+        with pytest.warns(UserWarning) if (round_idx == 0) and (
+            embedding_net_device != training_device
+        ) else nullcontext():
+            density_estimator_train = density_estimator_append.train(max_num_epochs=2)
+
         posterior = inference.build_posterior(density_estimator_train)
         proposal = posterior.set_default_x(x_o)
         theta = proposal.sample((samples_per_round,))